A huge country with a huge dark tourism potential – but still a tough nut to crack, as far as organizing travel in these parts is concerned. But it is worth the effort, as long as you are prepared to deal with some typical post-Soviet inconveniences and uncertainties and start planning ahead early.
UPDATE 2017: things have got a bit easier in the meantime. Good news #1: citizens of 45 nations can now enter Kazakhstan visa-free for up to 30 days.
Good news #2: Koryo Tours now offer a splendid package tour to Kazakhstan that includes many of the places covered here, most importantly the Polygon and Kurchatov, which makes these difficult to reach places much easer to get to. See the sponsored page for Koryo Tours.
Kazakhstan's enormous land mass stretches from the edge of Eastern Europe deep into Central Asia, formerly the largest of the Soviet republics after Russia. Everything is remote in this vast area that is mostly empty steppe; but for the really determined dark traveller it includes some extra-special destinations, in particular the Polygon, the former nuclear test site of Semipalatinsk in the east (with its former HQ Kurchatov and the nearby ghost town of Chagan at an ex-air-force base) – and the dying Aral Sea in the south-west. In addition, the former Soviet, now international "Space Centre" of Baikonur, east of the Aral Sea, is a unique attraction. It also used to be the testing ground for Soviet nuclear ICBMs during the Cold War. And then there are the sites of former Soviet (Stalinist) gulags, of course.
In addition, the old industrial parts (e.g. around Ekibastuz and Karaganda) as well as aspects of modern Kazakhstan (such as the cult of personality revolving around the president) can be of wider interest for dark tourists too.  
The tourism infrastructure outside the main cities of Almaty and Astana isn't as developed as it is in many other countries, so visiting some of the more extreme dark destinations listed here is more of an expedition than a leisurely trip.
There are, however, tour operators that offer various packages and tailor-made tours in relative comfort, some even including trips to truly exotic locations such as the Polygon of Semipalatinsk (see below).

    Semipalatinsk/Semey (with the anatomical museum)
            + the Polygon nuclear test site area
    Kurchatov (with the Museum of the Semipalatinsk Test Site)
    Chagan ghost town and former strategic bomber base
    Aralsk on the Aral Sea
    Baikonur Cosmodrome
    KarLag & Spassk gulag memorials
    ALZHIR memorial museum for the victims of repression
    Astana (Museum of the First President, the President's Cultural Centre)
    Karaganda (with the Ecomuseum)
    Ekibastuz (with regional museum, Bogatyr open-cast coal mining pit)
    Aksu with its socialist sculpture park
    Temirtau industrial mammoth
    Pavlodar, provincial capital of Soviet/Russian character
Officially, tourism is one of the things that the country wants to promote, but especially from a dark tourism perspective that doesn't necessarily mean much. Planning a trip to take in the sites listed here is an effort and takes time and stamina. But it is doable.
UPDATE January 2017: Kazakhstan has suspended the visa requirement for citizens of 45 countries (including almost all European countries, North America and some Asian countries such as, UAE, Singapore, Malaysia, Japan and South Korea) for stays of up to 30 days!
If you do have to apply for a visa, you need to submit not only the usual filled-in form with a passport photo and a passport that is valid for a minimum of 6 months after the date of travel, you also have to hand in a covering letter detailing at least your first accommodation's address (you may also have to provide proof of booking) and preferably also those of any tour operators used. The fee I had to pay for a one-month single entry tourist visa in 2011 was 30 EUR. Other visa types obviously cost more. 
Getting to Kazakhstan is easiest by plane, most likely flying to either Almaty or Astana, whose international airports are served by a number of airlines connecting to various European and Asian cities. Apart from the national carrier Air Astana, you can fly with e.g. Lufthansa, Turkish Airlines, Austrian, Ukrainian and various Russian carriers. Fares can be quite steep, but it pays off to start shopping around early and to have some flexibility with flight dates in order to find a good deal (I managed to get a very good one with Ukrainian, via Kiev).
In theory, you can also travel to Kazakhstan by train: from Europe e.g. via Berlin and Moscow; and train lines also extend into the southern neighbouring countries. Needless to say, going there by train means an extra bureaucratic effort and takes much longer.
Within the country, however, travelling by train is a very good option, albeit not necessarily a fast one, but cheap. An exception is the high-speed train between Almaty and Astana, which is both fast and expensive (it costs nearly as much as an internal flight). Where there are no train lines, buses provide an alternative. If you have to travel longer distances within the country (which is very likely, given its enormous size), internal flights may be the only viable option if you're pressed for time. They are comparatively inexpensive.
Many of the places listed above can best be visited as part of an organized tour, and such operators often use their own vehicles, minibuses or even 4x4s (which in some areas are indispensable anyway). Operators include the Dutch outfit Kazachstan Tours (sponsored page here), and - new from 2018! - Koryo Tours (sponsored page here)
Getting behind the wheel and driving yourself in Kazakhstan is only recommended for the adventurous. That said, though, I was positively surprised at how disciplined drivers are in the cities with regard to stopping for pedestrians (especially compared to other post-Soviet countries such as Turkmenistan or Georgia).
Accommodation in Kazakhstan can be quite expensive. At the upscale end prices reach at least western levels and are often even higher. There is precious little on offer at the budget end of the scale, but the mid range is OK. However, you may have to get used to some ex-Soviet hotel quirkiness such as a grim 'dizhurnaya' lady on every floor controlling access to her realm's rooms. Especially in more provincial towns you also have to be prepared for power cuts and/or periods during which there is no hot water or even no running water at all. And in extremely off-the-beaten-track places such as Aralsk, things can get pretty rough.  
In terms of food & drink, Kazakhstan, like the whole of Central Asia, is, from a semi-vegetarian perspective such as mine, well, a wasteland. A nightmare even. The Kazakhs belong to most carnivorous peoples on Earth and a meal not containing meat is more or less unthinkable to them. So it is hard to get by as a veggie, at least with Kazakh cuisine. Often you will be forced to fall back on international standards such as the ubiquitous pizza. But even meat-lovers will find some Kazakh specialities difficult to swallow or even nibble at. Roast sheep heads are an example – and these even play a prestigious role and as such are often offered to "special guests" first (and foreigners invited into a Kazakh home are almost by default special guests).
On the drinks front, some local standards are also an acquired taste, such as 'kumiss', fermented mare's milk. For me, the process of trying to acquire that particular taste went kind-of in reverse: the first sip was alright, the second still so-so, and thereafter I quickly went off the stuff. Strong, sweetened black tea is common too, just like everywhere in the Orient. Although Kazakhstan is predominantly Muslim, beer is readily available everywhere, as is vodka – the latter a legacy of the Tsarist and Soviet eras, of course. There's also brandy, called Konyak, which is perfectly palatable, though not exceptional. Wine is only available in the form of heavily overpriced imported bottles (well, there is also Kazakh-made wine, which I tried once, and only once … the less said about that the better …).  
One very positive surprise I was not at all prepared for when I went, in August 2011, was the Kazakhs' considerateness with regard to smoking. This may be a more recent development, as even bars and restaurants still described in the 2008 Bradt guide for Kazakhstan as "smoke-filled" now follow a blanket ban on smoking in public places. And this is indeed adhered to with astounding discipline. Even chain-smokers go outside for every single one of their puffs – which means you can see lots of tables laden with food and drink but no diners present, because they're outdoors smoking most of the time. It's a maximal contrast to the anarchy with regard to no-smoking rules encountered in countries such as Albania or Montenegro, where everyone smokes everywhere despite the no-smoking signs. Not in Kazakhstan. Here it appears to work in a highly commendable fashion. And I found that quite a relief, I must say.
Language-wise it is a huge advantage if you know at least some Russian, which is the de facto lingua franca in Kazakhstan. Officially, Kazakh is the first national language, but even many Kazakhs do not speak it fluently. It's a Turkic language and as such is similar to Turkish, Uzbek, Azeri or Turkmen. Signs in the streets are usually bilingual, Kazakh and Russian – occasionally you can even see English as a third language. English may get you only so far in the capital and cosmopolitan Almaty, but it will be practically of no use whatsoever elsewhere. However, tour operators covering the more remote places listed above will provide English-speaking guides or interpreters.   
Speaking of operators and logistics … My trip to Kazakhstan in late summer 2011 took the longest planning of any of the countless journeys I have undertaken in my travelling life. I started researching my options well over a year in advance and got down to serious planning about 11 months ahead of the prospective start date of the trip. It soon became apparent that this was not going to be a breeze. Of the various operators I found offering tours to the more remote places I wanted to see (Semipalatinsk Test Site, the Aral Sea, Baikonur) some never responded at all, others did for a while but then suddenly fell silent never to respond to anything again. These were not just companies in Kazakhstan itself, but also e.g. a British-based one! (I won't name names here – but if you contact me I can specify who these companies were.) Of the companies I did eventually end up using, a couple also underwent phases of, well, inconsistent communication, shall we say (partly due to a change in the personnel involved). And even with everything supposedly finalized and ready to go, there were more last-minute changes and vagaries of various kinds. But in the end, sometimes with a bit of luck, it all more or less worked out OK … with the massive exception of Baikonur (last-minute tour cancellation due to a series of accidents, so not the tour operator's fault – and I was reimbursed for the full price of the tour). It was a great disappointment – but could not be helped.
While, generally speaking, communication proved the hardest nut to crack with most Kazakhstan operators, I have to point out the shining exception to that generalization, which is, not least because of the excellent communication, my recommended first port of call for anyone thinking about visiting the places outlined on these pages: it is a Dutch company (but also operating in English and German) trading under the alternate English and Dutch names Kazakhstan Tours or "Kazachstan Reizen" – see the sponsored page here. The advantage of using this company, apart from their general competence, is also that you can keep it all in one pair of hands, as it were, rather than having to deal directly with different local companies in the various locations (like I did).
UPDATE: from 2018 the trusted operator Koryo Tours (mainly a specialsit in North Korea, hence the name) also offers an exciting package programme for Kazakhstan that includes teh otherwise hard-to-get-to Polygon, Kurchatov and Chagan. Check their sponsored page here.
Still, you would have to get into gear with the planning in good time all the same, especially if you are thinking of including "sensitive" places such as Baikonur or the Polygon/Kurchatov, where special permits need to be secured many weeks, if not months, in advance!
In short: Kazakhstan is not an "easy" destination, but for those willing to invest the requisite time, nerves and money it offers some top-notch dark attractions that are pretty much unique in the world.
Oh yeah, one final remark – as it still seems to be necessary to make this clear: the real Kazakhstan has nothing whatsoever to do with that "portrayed" in the "Borat" film by the British comedian Sacha Baron Cohen (and occasionally also in his series "Ali G"). You can't even say "portray", really, as not one second of this was actually shot in Kazakhstan. (Instead a poor rural village in Romania had to stand in as a pseudo film set – even though that place didn't actually bear any significant resemblance to Kazakhstan either.) But that's not the point anyway. Cohen just chose as the country of origin for his hapless "reporter" character a place that he assumed (probably rightly so) no one in the West would know anything about – so that he would have free rein with his fictional mayhem. After all, his comical attacks and exaggerated exploitation of prejudices are directed not against Kazakhstan but the West, in particular the USA (with some spot-on success here and there). Whatever, and whether or not you like the comedy style of the film (personally I'm not really sure about it either way), one just has to remember that none of this has anything to do with the real Kazakhstan. End of.   
  • Kazakhstan 01 - pretty flagKazakhstan 01 - pretty flag
  • Kazakhstan 02 - cliche horses in the steppeKazakhstan 02 - cliche horses in the steppe
  • Kazakhstan 03 - Khodja Ahmed Yassaui mausoleum in TurkestanKazakhstan 03 - Khodja Ahmed Yassaui mausoleum in Turkestan
  • Kazakhstan 04 - Silk Road grandeurKazakhstan 04 - Silk Road grandeur
  • Kazakhstan 05 - some lesser ancient cultural relicKazakhstan 05 - some lesser ancient cultural relic
  • Kazakhstan 06 - Soviet relics with MiG on a plinthKazakhstan 06 - Soviet relics with MiG on a plinth
  • Kazakhstan 07 - rural lifeKazakhstan 07 - rural life
  • Kazakhstan 08 - bringing home the hayKazakhstan 08 - bringing home the hay
  • Kazakhstan 09 - very Russian-looking wooden houseKazakhstan 09 - very Russian-looking wooden house
  • Kazakhstan 10 - rural facilitiesKazakhstan 10 - rural facilities
  • Kazakhstan 11 - not quite totally flat steppeKazakhstan 11 - not quite totally flat steppe
  • Kazakhstan 12 - there are even forested hillsKazakhstan 12 - there are even forested hills
  • Kazakhstan 13 - lots of train trafficKazakhstan 13 - lots of train traffic
  • Kazakhstan 14 - no straightforward trafficKazakhstan 14 - no straightforward traffic
  • Kazakhstan 15 - Kazakh militaryKazakhstan 15 - Kazakh military
  • Kazakhstan 16 - big Nazarbayev is watching youKazakhstan 16 - big Nazarbayev is watching you
  • Kazakhstan 17 - red sunset sky with black pollutionKazakhstan 17 - red sunset sky with black pollution
  • Kazakhstan 18 - a gulag for chickensKazakhstan 18 - a gulag for chickens
  • Kazakhstan 19 - not just vegetarians are in hell hereKazakhstan 19 - not just vegetarians are in hell here
  • Kazakhstan 20 - horse meat specialityKazakhstan 20 - horse meat speciality
  • Kazakhstan 21 - TengeKazakhstan 21 - Tenge
  • Kazakhstan 22 - maximum clicheKazakhstan 22 - maximum cliche
  • Kazakhstan 23 - saiga antelopes once roamed the steppe in their millionsKazakhstan 23 - saiga antelopes once roamed the steppe in their millions

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