The Moscow Virtuosi ensemble was founded by violist and conductor Yuri Bashmet in 1986. In 1992 the ensemble was completely revamped, taking in graduates and postgraduates of the Moscow Conservatoire. It made its debut on 19 May 1992 at the Great Hall of the Moscow Conservatoire. Two days later it gave its first performance abroad, at the Salle Pleyel in Paris.

The ensemble has given concerts at Carnegie Hall in New York, the Concertgebouw in Amsterdam, Suntory Hall in Tokyo, the Barbican Hall in London, the Tivoli in Copenhagen, the Berliner Philharmoniker, the Sydney Opera House, the Musikverein in Vienna, the Accademia Nazionale di Santa Cecilia in Rome and the Great Hall of the Moscow Conservatoire.

The ensemble takes part in the Proms concerts at the Royal Albert Hall in London, the Mstislav Rostropovich Festival in Evian, Sony-Classical sponsored concerts at the Théâtre des Champs Élysées, Semaines musicales de Tours, Elba Isola Musicale d'Europa, December Evenings, A Dedication to Oleg Kagan, Prestige de la Musique at the Salle Pleyel, the World Chamber Orchestras Festival in Omsk and festivals in Ravenna, Montreux, Bath, Sydney, Qabala and Moscow.
Between 2008 and 2014 the ensemble took part in the Sochi Winter Festival, of which Yuri Bashmet is Artistic Director. The Moscow Soloists take part in Bashmet’s Moscow International Viola Competition and his festivals in Yaroslavl, Khabarovsk, Rostov-on-Don, Minsk and the Seychelles. In January 2013 the ensemble appeared at a festival commemorating the maestro’s sixtieth birthday.

The ensemble’s concerts are frequently broadcast and recorded by the world’s leading broadcasting companies, among them the BBC, Bayerische Rundfunk, Radio France and NHK. The orchestra has performed with Sviatoslav Richter, Mstislav Rostropovich, Natalia Gutman, Viktor Tretyakov, Gidon Kremer, Maxim Vengerov, Vadim Repin, Sarah Chang, Shlomo Mintz, Barbara Hendricks, James Galway, Lynn Harrell, Mario Brunello, Thomas Quasthoff, Anna Netrebko, Olga Borodina Jessye Norman and Yefim Bronfman.

The repertoire of the Moscow Soloists includes over three hundred and fifty masterpieces of world classics and rarely performed works, ranging from Bach and Mozart to Schnittke and Denisov as well as music by Kancheli, Gubaidulina and other contemporary composers.

In 2008 the Moscow Soloists received a Grammy award for its recording of music by Stravinsky and Prokofiev. In 1994, 2006 and 2009 the ensemble was a Grammy award nominee.
In 2007, to mark fifteen years since it was founded, the ensemble undertook a tour of Russia, during which it gave forty-two concerts in thirty-nine towns and cities. In Ufa the musicians performed their one thousandth concert, while their concert in Severomorsk took place on the cruiser “Peter the Great”. The ensemble undertook an even larger tour to mark its twentieth anniversary, giving over eighty concerts in thirty countries.

In the autumn of 2009 the Moscow Soloists undertook a tour of Russian towns and cities during which they performed on unique instruments crafted by Antonio Stradivari from the Russian State Collection of Prized Musical Instruments. In the 2013–2014 season the ensemble’s musicians undertook a similar tour of Europe’s capital cities.

In 2014 the ensemble took part in the cultural programme of the Winter Olympic Games in Sochi.