Aleksandr Mikhailovich Lyapunov (Russian: Алекса́ндр Миха́йлович Ляпуно́в; June 6 [O.S. May 25] 1857 – November 3, 1918) was a Russian mathematician, mechanician and physicist. His surname is sometimes romanized as Ljapunov, Liapunov or Ljapunow.

Lyapunov is known for his development of the stability theory of a dynamical system, as well as for his many contributions to mathematical physics and probability theory.

Lyapunov was born in Yaroslavl, Russian Empire. His father Mikhail Vasilyevich Lyapunov (1820–1868) was an astronomer and a head of the Demidovski lyceum. His brother, Sergei Lyapunov, was a gifted composer and pianist. In 1863, M. V. Lyapunov retired from his scientific career and relocated his family to his wife's estate at Bolobonov, in the Simbirsk province (now Ulyanovsk Oblast). After the death of his father in 1868, Aleksandr Lyapunov was educated by his uncle R. M. Sechenov, brother of the physiologist Ivan Mikhailovich Sechenov. At his uncle's family, Lyapunov studied with his distant cousin Natalia Rafailovna, who became his wife in 1886. In 1870, his mother moved with her sons to Nizhny Novgorod, where he started the third class of the gymnasium. He graduated from the gymnasium with distinction in 1876.

Sergei Mikhailovich Lyapunov (Russian: Серге́й Миха́йлович Ляпуно́в; November 30 [O.S. November 18] , 1859 – November 8, 1924) was a Russian composer and pianist.

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).

He graduated in 1883, more attracted by the nationalist elements in music of the New Russian School than by the more cosmopolitan approach of Tchaikovsky and Taneyev. He went to St. Petersburg in 1885 to seek Mily Balakirev, becoming the most important member of Balakirev's latter-day circle. Balakirev, who had himself been born and bred in Nizhny Novgorod, took Lyapunov under his wing, and oversaw his early compositions as closely as he had done with the members of his circle during the 1860s, now known as The Five. Balakirev's influence remained the dominant influence in his creative life.