Anatoly Kuksov was born into a large family in Lugansk, where was one of five boys out of six children, and was a pupil of the Luhansk sporting school “Trudovye Rezervy”. Throughout his career he played for just one club, Zarya (Luhansk). Between 1966-85, Kuksov played 517 matches and scored 89 goals. He had offers to move to leading USSR teams, but remained loyal to Zarya, and in 1972 helped them to become USSR champions. That same year he played eight matches for the national team and took part in five games at the 1972 München Olympics.
After his playing career Kuksov worked as the head coach of various football teams in the Lugansk, Donetsk, and Zaporozhye regions of Ukraine. In 2015, he headed the national team of the Lugansk People’s Republic, and in March that year, they played their only international match against the national team of Abkhazia, who won 1-0. However, the match was unofficial as the two nations were not recognised by FIFA. In July 2018, the Lugansk People’s Republic post office issued a stamp with a portrait of Anatoly Kuksov on it.