Bykova was born in Azov, Rostov, Russia and first came to international attention at the 1980 Olympic Games in Moscow, where she finished ninth in the final with 1.88 meters. Six weeks later she won the Soviet championship with a jump 1.97 meters. At the 1981 World Cup in Rome, she cleared 1.96 m to finish second to West Germany's Ulrike Meyfarth, who set a new world record with a jump of 2.02 meters. At the 1982 European championship in Athens, Bykova cleared 1.97 m to again finish second to Meyfarth. Then at the 1983 European Indoor Championship in Budapest, she cleared 2.03 meters to win the gold medal and set a new world indoor record.

At the 1983 World Championships in Helsinki, Bykova and Meyfarth were the only to jumpers to clear 1.99 meters, but only Bykova could manage the next height of 2.01 meters, to become the inaugural world champion. The next meeting between the two came one month later at the European Cup in Crystal Palace in London. This time Meyfarth set a new world record by jumping over 2.03 meters, but only a few minutes later the Russian jumped over the same height to equal the world record, however she had needed one more attempt than the German and had to settle yet again for second place. Only four days later the two met again, this time in Pisa. This time though, Bykova came out on top with a new world record of 2.04 meters.

In June 1984, Bykova once again broke the world record with a clearance of 2.05 metres. This height would remain her lifetime best. The record would stand for only a month, as Bulgaria's Lyudmila Andonova cleared 2.07 metres in July. Bykova was prevented from competing at the 1984 Olympic Games in Los Angeles due to the Soviet boycott.

In 1987, Bykova won a silver medal at the World Championships in Rome, with a clearance of 2.04 metres. The winner was Stefka Kostadinova of Bulgaria, with a new world record of 2.09 metres. At the 1988 Olympic Games in Seoul, Bykova won the bronze medal with 1.99 metres, behind the American gold medal winner Louise Ritter and the silver medallist Kostadinova.

Bykova received a three-month ban when she tested positive for the drug ephedrine at the Goodwill Games in 1990, and missed the European Championships held later that year.

• List of sportspeople sanctioned for doping offences

• Tamara Bykova's profile in the Modern Museum of Sports features her photos and photos of her awards and decorations (in Russian)