Leonov was born on 30 May 1934 in Listvyanka, West Siberian Krai, Russian SFSR. His grandfather had been forced to relocate to Siberia for his role in the 1905 Russian Revolution. Alexei was the eighth of nine surviving children born to Yevdokia née Sotnikova and Arkhip. His father was an electrician and miner.

In 1936, his father was arrested and declared an “enemy of the people”. Leonov wrote in his autobiography: “He was not alone: many were being arrested. It was part of a conscientious drive by the authorities to eradicate anyone who showed too much independence or strength of character. These were the years of Stalin’s purges. Many disappeared into remote gulags and were never seen again.” The family moved in with one of his married sisters in Kemerovo. His father rejoined the family in Kemerovo after he was released. He was compensated for his wrongful imprisonment. Leonov used art as a way to provide more food for the family. He began his art career by drawing flowers on ovens and later painted landscapes on canvasses.

The Soviet government encouraged its citizens to move to Soviet-occupied Prussia, so in 1948 his family relocated to Kaliningrad. Leonov graduated from secondary school (No. 21) in 1953. He applied to the Academy of Arts in Riga, Latvia, but decided not to attend due to the high tuition costs. Leonov decided to join a Ukrainian preparatory flying school in Kremenchug; he made his first solo flight in May 1955. While indulging in his passion for art by studying part-time in Riga, Leonov started an advanced two-year course to become a fighter pilot at the Chuguev Higher Air Force Pilots School in the Ukrainian SSR.

On 30 October 1957, Leonov graduated with an honours degree and was commissioned a lieutenant in the 113th Parachute Aviation Regiment, part of the 10th Engineering Aviation Division of the 69th Air Army in Kyiv. On 13 December 1959, he married Svetlana Pavlovna Dozenko a day before he moved to East Germany to his new assignment with the 294th Reconnaissance Regiment of the 24th Air Army.

He was one of the 20 Soviet Air Forces pilots selected to be part of the first cosmonaut training group in 1960. As most cosmonauts, Leonov was a member of the Communist Party of the Soviet Union. His walk in space was originally to have taken place on the Voskhod 1 mission, but this was cancelled, and the historic event happened on the Voskhod 2 flight instead. He was outside the spacecraft for 12 minutes and nine seconds on 18 March 1965, connected to the craft by a 4.8-metre (16 ft) tether. At the end of the spacewalk, Leonov’s spacesuit had inflated in the vacuum of space to the point where he could not re-enter the airlock. He opened a valve to allow some of the suit’s pressure to bleed off and was barely able to get back inside the capsule. Leonov had spent eighteen months undergoing weightlessness training for the mission.

Leonov was an accomplished artist whose published books include albums of his artistic works and works he did in collaboration with his friend Andrei Sokolov. Leonov took coloured pencils and paper into space, where he sketched the Earth and drew portraits of the Apollo astronauts who flew with him during the 1975 Apollo–Soyuz Test Project. Arthur C. Clarke wrote in his notes to 2010: Odyssey Two that, after a 1968 screening of 2001: A Space Odyssey, Leonov pointed out to him that the alignment of the Moon, Earth, and Sun shown in the opening is essentially the same as that in Leonov’s 1967 painting Near the Moon, although the painting’s diagonal framing of the scene was not replicated in the film. Clarke kept an autographed sketch of this painting—which Leonov made after the screening—hanging on his office wall. Clarke dedicated 2010: Odyssey Two to Leonov and Soviet physicist Andrei Sakharov; and the fictional spaceship in the book is named Cosmonaut Alexei Leonov.

In 1968, Leonov was selected to be commander of a circumlunar Soyuz 7K-L1 flight. This was cancelled because of delays in achieving a reliable circumlunar flight (only the later Zond 7 and Zond 8 members of the programme were successful) and the Apollo 8 mission had already achieved that step in the Space Race. He was also selected to be the first Soviet person to land on the Moon, aboard the LOK/N1 spacecraft. This project was also cancelled. (The design required a spacewalk between lunar vehicles, something that contributed to his selection.) Leonov was to have been commander of the 1971 Soyuz 11 mission to Salyut 1, the first crewed space station, but his crew was replaced with the backup after one of the members, cosmonaut Valery Kubasov, was suspected to have contracted tuberculosis (the other member was Pyotr Kolodin).

From 1976 to 1982, Leonov was the commander of the cosmonaut team (“Chief Cosmonaut”) and deputy director of the Yuri Gagarin Cosmonaut Training Center, where he oversaw crew training. He also edited the cosmonaut newsletter Neptune. He retired in 1992.

In 2001, he was a vice president of Moscow-based Alfa-Bank and an adviser to the first deputy of the Board.

In 2004, Leonov and former American astronaut David Scott began work on a dual memoir covering the history of the Space Race between the United States and the Soviet Union. Titled Two Sides of the Moon: Our Story of the Cold War Space Race, it was published in 2006. Neil Armstrong and Tom Hanks both wrote introductions to the book.

Leonov died on 11 October 2019 after a long illness in Moscow. His funeral took place on 15 October. He was 85 and the last living member of the five cosmonauts in the Voskhod programme. He was survived by his wife Svetlana Dozenko, daughter Oksana, and two grandchildren; his other daughter, Viktoria, died in 1996.