Alexei Nikolaevich Krylov was born on August 3 O.S., 1863 to the family of an Army Artillery officer in a village of the Simbirsk Gubernia in Russia. Nikolay Krylov's parents were rather poor, but he received a free education as the son of an army veteran.

Krylov entered a Naval College (Морское училище) in 1878 and finished with distinction in 1884. There he did his first scientific work with Ivan de Collong about Deviation of magnetic compasses. The theory of magnetic and gyro-compasses fascinated him all his life; he later published important works related to the dynamics of the magnetic compass and proposed the dromoscope, a device that would automatically calculate the deviation of a compass. He also was a pioneer of the gyrocompass, being the first to create a full theory of it.

After spending several years at the Main Hydrographic Administration and at a ship building plant (French - Russian shipbuilding company), in 1888 he continued his study in the Naval Academy of Saint Petersburg. He was a talented and promising student and after graduating ahead of schedule from the Academy in 1890, stayed on as Mathematics and Ship theory lecturer.

Fame came to him in the 1890s, when his pioneering Theory of oscillating motions of the ship, significantly extending William Froude's rolling theory, became internationally known. This was the first comprehensive theoretical study in the field. In 1898 Krylov received a Gold Medal from the Royal Institution of Naval Architects, the first time the prize was awarded to a foreigner. He also created a theory of damping of ship rolling and pitching, and the was first to propose gyroscopic damping which now is the most common way of damping the roll.

After 1900 Krylov actively collaborated with Stepan Makarov, admiral and maritime scientist, working on the ship floodability problem. The results of this work soon became classic and are used today worldwide. Years later, Krylov wrote about of the early ideas of Makarov to fight the heel of a sinking ship by flooding its undamaged compartments: This appeared to be such a great nonsense [to the naval officials] that it took 35 years… to convince [them] that the ideas of the 22-year-old Makarov are of great practical value.

Krylov was well known for his sharp tongue and quick wits. His put downs to government and Duma officials were legendary. As a capable naval consultant, he claimed that his advice saved the government more than the cost of a dreadnaught.

In 1917 he became CEO of the Russian society for shipbuilding and trade (Русское общество пароходостроительства и торговли). After the October Revolution he transferred all his ships to Soviet government and continued to work for the Russian Navy. In 1921 he went to London to re-establish scientific contacts, working there as a representative of the Soviet government. In 1927 he returned to the Soviet Union.

Krylov wrote about 300 papers and books. They span a wide range of topics, including shipbuilding, magnetism, artillery, mathematics, astronomy, and geodesy. His floodability tables have been used worldwide. Of note are his works in hydrodynamics including theory of ships moving in shallow water (he was the first to explain and calculate the significant increase of hydrodynamic resistance in shallow water) and the theory of solitons. In 1904 he built the first machine in Russia for integrating ODEs.

In 1931 he published a paper on what is now called the Krylov subspace and Krylov subspace methods. The paper deals with eigenvalue problems, namely, with computation of the characteristic polynomial coefficients of a given matrix. Krylov was concerned with efficient computations and, as a real computational scientist, he counts the work as number of separate numerical multiplications — something not very typical for a 1931 mathematical paper. Krylov begins with a careful comparison of the existing methods that include the worst-case scenario estimate of the computational work in the Jacobi method. Later, he presents his own method which is superior to the known methods of that time and is still widely used.

Krylov also published the first Russian translation of Isaac Newton, Philosophiæ Naturalis Principia Mathematica (1915).

Alexei Nikolaevich Krylov died in Saint Petersburg (by that time Leningrad) on October 26, 1945, shortly after the end of the World War II. He is buried in the Volkovo Cemetery, not far from the physiologist Ivan Pavlov and the chemist Dmitri Mendeleev. He was awarded the Stalin Prize (1941), three Orders of Lenin, Hero of Socialist Labor (1943), and was an academician of the Russian Academy of Sciences (after 1916). The crater Krylov on the Moon is named after him.

In one of his autobiographical papers, Krylov describes his activity as 'shipbuilding, i.e. application of Mathematics to various Maritime problems.'

Krylov married his second cousin Elisaveta Dmitrievna Dranitsyna. His daughter Anna married the famous physicist Pyotr Kapitsa. Alexei Krylov was very close to his son-in-law.