NASA's space shuttle programme was started when the Cold War began to grow hot: the first flights took place in the era of Cruise missiles, Protect and Survive, the doctrine of a winnable nuclear war uncovered by Robert Sheer's With Enough Shovels, The Day After, and Frankie Goes To Hollywood's 'Two Tribes'. The Soviet authorities realised that the shuttle had serious military uses, and decided to start their own programme. The spacecraft in the image above, Buran, is the only Soviet shuttle to have reached orbit. Launched in November 1988, it was unmanned, completed two orbits of the Earth, and landed under automatic guidance. There's more information about it here and here.
Within a year, history had overtaken the Buran programme. The Berlin Wall fell, the Soviet Union collapsed, and the authorities realised that their space shuttle was an expensive dead end which could no longer be justified, and shut it down (the USA took somewhat longer to come to the same conclusion). Four shuttles were under construction at the time. One, nicknamed Ptichka (Little Bird), is stored in the Baikonur Cosmodrome alongside a non-flying prototype; another, Baikal, is parked on an airfield; the other two have been partially or completely dismantled. Two prototypes are on public display: one in Gorky Park, Moscow; the other in the Technik Museum Speyer, Germany.
As for Buran, the only Soviet shuttle to have orbited the Earth, it was destroyed when the roof of the hangar in which it is was being stored at Baikonur collapsed. An ignominious end to the avatar of an alternate history which might have intensified the cold war in low Earth orbit, or which might have seen two kinds of space shuttles servicing the International Space Station, but which otherwise, let's face it, probably wouldn't have been very different to our own history.