Alexei Lubimov studied at the Moscow Conservatory with Heinrich Neuhaus and Lew Naumov. As early as 1968 Lubimov gave concerts in Soviet Russia of contemporary western composers. Although criticized by the Soviet government, this did not stop him and in 1976 he founded the Baroque Quartet and the Academy of Arts in Moscow.
He has several world tours to his name, for example 1991 and 2010 in the USA, but he still performs mainly in Russia and Europe. He has performed several times in the Dutch Concertgebouw.
In recent years he has given concerts with the London Philharmonic Orchestra, the Russian National Orchestra and the Tonkünstlerorchester. He has made tours as a soloist, but also with the Haydn Sinfonietta and with the Camerata Salzburg by Sir Roger Norrington. In 2011 he toured with the Budapest Festival Orchestra.
Lubimov was a jury member for the International Harpsichord and Piano Forum Contest in Bruges, as part of the Musica Antiqua Festival for the years 2004, 2007 and 2010. He taught at the Mozarteum in Salzburg.