Bareev's tournament wins span from different international stages. He won at Dortmund in 1992 and Wijk aan Zee in 2002. In the same year, he emerged victorious at the Russian Championships and the Essen rapid tournament. Besides, he reached the semifinals in the FIDE World Championships in 1999 and 2000.
Bareev also provided essential contributions to Russian team victories in numerous Chess Olympiads. In 1992, he was part of the Russian team that won the gold medal in Manila, Philippines. He competed in the Olympiad another seven times, with the team bagging the gold on five occasions.
In 2015, Bareev switched federations from Russia to Canada, and since then, he represented Canada in different tournaments. Among his recent tournaments is the 2017 summer event Pro League for the Montreal Chess Brahs team, which became the 2017 PRO Chess champions.
Apart from his skills in the board game, Bareev has a co-written book called From London to Elista, which provides an insider's view of the World Chess Championship matches disputed in 2006. The book is highly acclaimed for its anecdotes, analysis, and honest comments about top-level chess.