Lyapunov was born in Yaroslavl in 1859. After the death of his father, Mikhail Lyapunov, when he was about eight, Sergei, his mother, and his two brothers (one of them was Aleksandr Lyapunov, later a notable mathematician) went to live in the larger town of Nizhny Novgorod. There he attended the grammar school along with classes of the newly formed local branch of the Russian Musical Society. On the recommendation of Nikolai Rubinstein, the Director of the Moscow Conservatory of Music, he enrolled in that institution in 1878. His main teachers were Karl Klindworth (piano; a former pupil of Franz Liszt), and Sergei Taneyev (composition; a former pupil of Pyotr Ilyich Tchaikovsky and his successor at the Conservatory).
In 1893, the Imperial Geographical Society commissioned Lyapunov, along with Balakirev and Anatoly Lyadov, to gather folksongs from the regions of Vologda, Vyatka (now Kirov) and Kostroma. They collected nearly 300 songs, which the society published in 1897. Lyapunov arranged 30 of these songs for voice and piano and used authentic folk songs in several of his compositions during the 1890s.
He succeeded Nikolai Rimsky-Korsakov as assistant director of music at the Imperial Chapel, became a director of the Free Music School, then its head, as well as a professor at the St. Petersburg Conservatory in 1911. After the Revolution he emigrated to Paris in 1923 and directed a school of music for Russian émigrés, but died of a heart attack the following year. For many years the official Soviet line was that Lyapunov had died during a concert tour of Paris, no acknowledgement being made of his voluntary exile.