In 1972, Bashmet purchased[citation needed] a 1758 viola made by Milanese luthier Paolo Testore, which he uses for his performances to date.

He has performed in leading concert halls the world over: in Europe, USA, Canada, Latin America, Australia, New Zealand and Japan.

Yuri Bashmet has performed under many noted conductors, including Rafael Kubelík, Mstislav Rostropovich, Seiji Ozawa, Valery Gergiev, Gennady Rozhdestvensky, Colin Davis, John Eliot Gardiner, Yehudi Menuhin, Charles Dutoit, Neville Marriner, Paul Sacher, Michael Tilson Thomas, Kurt Masur, Bernard Haitink, Kent Nagano, Simon Rattle, Yuri Temirkanov, and Nikolaus Harnoncourt.

In 1992, Bashmet reconstituted the orchestra featuring some of the most talented young musicians of Russia who are graduates and postgraduate students of the Moscow Conservatory.

Another CD, of the Thirteenth String Quartet by Shostakovich and the quintet by Brahms (Sony Classics), was acclaimed "Best Album of 1998" by The Strad magazine, and was nominated for a Grammy Award.

Bashmet has performed recitals, as well as his joint performances together with noted musicians including Mstislav Rostropovich, Gidon Kremer, Sviatoslav Richter, Isaac Stern, Anne-Sophie Mutter, Martha Argerich, Evgeny Kissin, Mischa Maisky, M. Portal, Shlomo Mintz, Oleg Kagan, Roberto Carnevale, Viktor Tretiakov, and Maria João Pires.

Yuri Bashmet has participated in many festivals all over the world: Tanglewood (USA), Tours, Bordeaux, Biarritz, Menton (France), Brussels, Kainuu, Mikkeli (Finland), Kroyt (Germany), Siena, Sorrento, Camerino, Streza (Italy), and "March Music Lias" (Ruse, Bulgaria).

He is the President of the Lionel Tertis International Viola Competition in Great Britain, the member of Jury for the violists' contests in Munich and "Maurice Vie" in Paris.

Yuri Bashmet has participated in many large-scale charity actions of the international importance, in Carnegie Hall (with Elton John and Stevie Wonder), in London (in the memory of Princess Diana); in the concerts, whose proceeds were transferred to the funds aimed to help the victims of the natural disasters in Armenia, Japan, and to the funds helping handicapped children.

This award was granted to Gidon Kremer, Thomas Quasthoff, Victor Tretiakov, Valery Gergiev, Anne-Sophie Mutter, Olga Borodina, Irina Antonova, Natalia Gutman, and Evgeny Kissin.

Yuri Bashmet leads national education projects, as a presenter and art director of the TV programs The Station of Dream and Music in the Museums of the World.

Afterwards the musicians of the orchestra decided to stay in France, a decision untenable for Bashmet himself, who was committed contractually to return to Russia.

Previous recipients include Igor Stravinsky, Leonard Bernstein, Benjamin Britten, Yehudi Menuhin, Isaac Stern, Arthur Rubinstein, Dmitri Shostakovich, Mstislav Rostropovich, Sviatoslav Richter, and Gidon Kremer.

In 1999, by the Act of the Minister of Culture of the Republic of France, Yuri Bashmet was granted the rank of the Officer of Arts and Literature.

In 2000, the Russian Biographic Society granted Yuri Bashmet the honorary title "The Man of the Year", in 2001 – "Zealot of the Enlightenment".