Why is Russia no longer dominating the world chess championships? USSR (Russia) used to dominate world chess but that dominance seems to have faded away; with the Russians losing both the World Chess Championships and the Olympiad. 
What is it that the Russians are not doing right that the USSR used to do, and/or what is it that the other countries did to catch up and surpass?
The Soviet Union used to dominate chess because it had state-sponsored schools that provided high-quality trainers. After the Soviet Union broke up, funding dried up, and many of these trainers emigrated. Naturally the competitors caught up.
Vladimir Kramnik discusses some of this in an interview he gave with chess.com.

Chess.com: Much has been written about the legendary Soviet chess schools. You attended the Botvinnik school in the 1980s. What was it like?
Vladimir Kramnik: People always have a bit of an exaggerated picture of the Soviet chess schools, this idea that there were these concentration camps for kids, working 25 hours a day on chess. No, it was just a two-week session, held twice a year, and even during these two weeks, it was not so terribly intense.
The main asset of the Soviet chess school was actually the high standard of trainers throughout the country. In a world without internet and little exchange of information, this knowledge was hidden away behind the iron curtain. The reason the balance of power started to change after the iron curtain came down was because many of these Russian trainers emigrated, allowing western players and Chinese players to learn from them and get closer to the Russians.
You were 14 when the cold war ended and the Soviet Union began to collapse. How did that affect you?
Of course there were some issues, but I was just a kid at the time, so it didn’t make that big an impact on me. I think it was a much more difficult time for the older generations. For me, there were plusses and minuses. The plusses were that I could finally start going abroad to play tournaments, of my own will. The minuses were that all the state support for chess started to disappear.
The years 1990-1992 were a very difficult time financially in Russia, and many talented players of my generation didn’t manage to improve quickly enough. They ran out of money for trainers, and they ended up stuck at the same level. I was quite lucky in that I benefited from the state support of young talents when I was small, and then I reached the top of the game very quickly. But initially, money was not always easy.