He was born on 2 March 1930. In 1954, he graduated from Moscow State University with a degree in biology. He pursued his career in his alma mater, where he was awarded a Ph.D. in biophysics in 1964 and published more than 60 papers. As a researcher, he spoke out against the pseudoscience known as Lysenkoism, which was dominant in the USSR at that time. At the end of the 1960s, he was forced to leave the university due to his involvement in opposition activities.

He started his opposition activity at the time when the writers Yuli Daniel and Andrei Sinyavsky were repressed in the mid-1960s in the USSR. Kovalyov, as an employee of the Institute of Biophysics at Moscow University, inspired the writing of several letters to the communist authorities in their defense.

In the 1960s, he joined the effort to establish structures aimed at defending human rights in the USSR, and because of this activity he experienced the first repressions: interrogations and searches. From the early 1970s, he participated in the distribution of self-published publications: "Chronicle of Current Events" and "Chronicle of the Catholic Church in Lithuania".

In 1975, he was sentenced to 7 years in a labor camp – followed by 3 years of internal exile in Kolyma. He completed his sentence in 1984 and after his release, he continued his opposition activity. In 1989, he became a member of the Moscow Helsinki Watch Group, and was also elected a member of the leadership of the "Memorial" organization. After the dissolution of the Soviet Union, he turned to official politics. Among others, he was a co-author of the "Declaration of Human and Civil Rights in Russia" in 1991 and the law "On the rehabilitation of victims of political repression".

After 2000, he was an uncompromising critic of both Vladimir Putin's domestic and foreign policies.

He was awarded with the Grand Cross of the Order of Merit of the Republic of Poland (2009). In the mid-1990s, he was nominated for the Nobel Peace Prize.