Solovyov, a fighter pilot who had served in the Soviet Far East, joined the Soviet cosmonaut squad as a trainee in 1976. He flew into space for the first time in 1988 as commander of the Soyuz TM-5 mission on a 10-day flight to the Mir space station with a Bulgarian “guest-cosmonaut,” Aleksandr Aleksandrov.

In the early 1990s, Solovyov flew two long-duration missions to Mir, the first (Soyuz TM-9) lasting 179 days in 1990 and the second (Soyuz TM-15) lasting 189 days in 1992–93. During both flights, Solovyov, with his crewmates, conducted a total of six space walks to carry out repairs.

Solovyov played a key role in the cooperative shuttle-Mir missions in the early 1990s. In 1995 he and his crewmate were delivered to Mir aboard the space shuttle Atlantis (mission STS-71) for a 75-day stay on the station. He conducted three more space walks during this flight.

Although he was tapped to be on the first crew to the International Space Station, Solovyov declined to serve under an American commander. Instead, he flew his last space mission on Soyuz TM-26 in 1997–98, spending 198 days aboard Mir. With a grand total of 651 days in space, he became one of the most experienced space travelers of all time.

Solovyov retired from the cosmonaut squad in 1999. He became the president of For the Good of the Fatherland, a national organization that recognizes the work of Russians devoted to national social and cultural development.