Musica Viva's origins go back to 1978, when violinist and conductor Viktor Kornachev founded a musical ensemble of nine players – all young and enthusiastic, and recent graduates of Moscow's musical academies. By 1988 the ensemble had grown into a fully-fledged orchestra, now under the direction of Alexander Rudin – who also gave the group its name, Musica Viva – Latin for 'living music'. 

Under Rudin's leadership the orchestra has become a significat musical ensemble, achieving the pinnacles of artistic endeavour, and ranked along Russia's finest orchestras. Musica Viva today is an all-round musical ensemble - which performs the widest possible range of compositions in all styles and genres with an assured confidence. The orchestra's thoughtful programming embraces seminal mainstream works alongside fascinating rarities. The orchestra is adept in multiple performance practices, always aiming to get as close as possible to the style the composer intended – often stripping away agglutinated layers of tired ideas to reveal the freshness of the masterpiece which lies beneath. 

The quintessence of the orchestra's projects has been a cycle at the Moscow Philharmonia called Masterpieces & Premieres – in which musical masterworks are heard in their original splendour, alongside musical rarities which are returned once more to their deserved place on the concert platform. The performance of neglected masterworks forms a major part of the orchestra's repertoire. The orchestra has given the first Russian performances of works by Handel, the sons of JS Bach, Cimarosa, Dittersdorf, Dussek, Pleyel, Triclire, Folkman, Kozlovsky, Fomin, Vielgorsky, Alyabiev, Degtyarev, and many others. The broad stylistic range of the orchestra enables its members to perform both rarely-performed music and the music of contemporary composers. Over many years the Musica Viva orchestra has performed music by V. Artyomov, A. Pärt, A. Sallinen, V. Silvestrov, T. Mansuryan and other living composers.

In recent years Musica Viva has increasingly been involved in large-scale projects – concert performances of operas and oratorios, featuring performances by leading foreign singers and conductors. Under the baton of Alexander Rudin the orchestra has given Masses by Haydn and his oratorios The Creation and The Seasons; Mozart's opera Idomeneo, Weber's Oberon, Beethoven's Fidelio, Schumann's Requiem, Vivaldi's oratorio Juditha Triumphans, CPE Bach's The Last Suffering of Our Lord, as well as Degtyarev's Minin & Pozharsky or the Relief of Moscow. In collaboration with the British maestro Christopher Moulds the orchestra has given the Russian premieres of Handel's operas Orlando and Ariodante, as well as his oratorio Hercules, in the Tchaikovsky Concert hall. The premieres in the 2015-2016 season have been a concert performance of Hasse's oratorio I Pellegrini al Sepolcro di Nostro Signore and Handel's operatic serenata Aci. Galatea e Polifemo, in its Italian version of 1708 - at the Tchaikovsky Concert Hall in Moscow. One of the most sparkling innovations which maestro Rudin has introduced with Musica Viva has been the ballet divertissement “Variations On A Rococo Theme” by Tchaikovsky – staged by Bolshoi Theatre ballerina and choreographer Marianna Ryzhkina - also at the Tchaikovsky Concert Hall. In continuation of grand projects of Musica viva orchestra in the same hall  in 2017 were performed "The Paul" Mendelssohn’s Oratorio with "Kantorei der Schlosskirche Weilburg" and soloists (Germany) and also was realized “Orfeo ed Euridice” by Ch-W Gluck in  Paris release 1774 (2019, theatrical format) .As a part of the Handel Festival, which took place in Moscow in the Tchaikovsky Concert Hall in September 2021 Musica viva performed the oratorio Theodora (conductor Alexander Rudin) and Psalm Dixit Dominus (conductor Andreas Spering) by Handel with an international cast of soloists and the “Intrada” vocal ensemble.

The degree of immersion in music of many epochs has almost reached the level of musical archaeology. This was the basis behind the concert cycle of “Silver Classics”. The repertoire of “Silver Classics” is music which was highly prized in its own era, yet which has somehow slipped through the chinks of today's concert repertoire. The performers in this cycle include a Young Artistes Program of soloists who have emerged from the ranks of European music competitions, as well as a “Cello Assembly” to which maestro Rudin invites his fellow cellists. 

The last three seasons from 2018 Musica viva realized circle of concerts “Parizien and London Symphonies by Haydn”, which took place in Chamber Hall of Moscow International Performing Arts center. The circle, interrupted due pandemia, will be continue in the season 2021-22. In the 2022-23 season in the “Joseph Haydn. Sturm und Drang”, in which the Orchestra will continue to open to the audience the Symphonies of the composer of the 70s of the 18th century.

Musica Viva regularly invites musicians of world status to join its performances. These have included Christopher Hogwood, Sir Roger Norrington, Vladimir Jurowsky, Andraš Adorian. Robert Levin, Andreas Staier, Eliso Virsaladze, Natalya Gutman, Ivan Monighetti, Nikolai Lugansky, Boris Berezovsky, Alexei Lubimov, Giuliano Carmingola, Isabelle Faust, Thomas Zettmaier, Christian Tetzlaff and leading operatic prima donnas Joyce di Donato, Annic Massis, Vivica Genaux. Deborah York, Susan Graham, Malena Ernman, Stéphanie d’ Oustrac, Hibla Gerzmava, Julia Lezhneva, and many others. World-famous choruses including Collegium Vocale and  State choir of Latvia have appeared with the orchestra.

Musica Viva makes continuous appearances in major international music festivals, including repeatedly took part in the famous "La folle journee" ("Crazy days") (Nantes, France). Since 2019, Musica viva has been an annual participant in the Crazy Days in Yekaterinburg festival, which has become the Russian satellite of the festival “La folle journee”. The orchestra has toured to Germany, France, the Netherlands, Italy, Spain, Belgium, Japan, Czechia, Slovenia, Poland, Finland, Turkey, India, China and Taiwan. Every year the orchestra gives regular concerts in many Russian cities. 

The orchestra has recorded over the twenty albums. The golden collection can be called the recordings of the orchestra with maestro A. Rudin as a soloist: Cello Concerts by J.-B. Tricklir (Cello Classics), Cello Concerto by N. Myaskovsky (Cello Classics), Cello Concertos by Hasse, K. F. E. Bach, Hertel (Chandos), received rave reviews from international critics. The Grieg Album introduces A. Rudin as not only a conductor and cellist, but also a master of orchestration. The disc contains chamber works by Edvard Grieg in the orchestral version of A. Rudin. Also of particular interest are discs from the works of A. Alyabyev, M. Glinka, Dvorak (Fuga Libera, Belgium), The Cherepnin Family (Olympia), as well as live albums - recordings of Beethoven's 3rd Symphony and Concerto for cello and orchestra by A. Kraft (Melody), Beethoven's 1st and 6th symphonies from the Big Hall of Moscow Conservatory (Fuga libera). In 2019, Naxos released the album of Symphonies by I. Stamitz; in 2020 Beethoven's Triple and Violin Concertos with Dmitry Sinkovsky (violin), Alexei Lyubimov (historical piano), Alexander Rudin (cello, conductor) (Note 1 music gmbh, Heidelberg). In 2022 Naxos releases a new album of works by A. Kraft and B. Romberg for cello duo (Alexander Rudin and Emin Martirosyan), with the participation of “Musica viva”.