Anatoly Kislyakov was born in 1961 in Novokuznetsk, in the Kemerovo region. In 1980, he graduated from the Moscow Choral School, today named AV Sveshnikov. In the same year, he entered the Moscow Conservatory in the choral conducting class of Professor Boris Grigorievich Tevlin. He combined his studies with work in the student choir.

After graduating from the Conservatory and serving in the country's Armed Forces, he worked in the Republic of Bulgaria with children's and mixed choir groups. Since 1996, at the invitation of VS Popov, he worked at the Academy of Choral Arts as a leader of study and concert groups, as well as a teacher of special choral disciplines.

In 2000, he started working in "The Grand Children's Choir of VGTRK", now named "The Grand Children's Choir after VS Popov". Since 2006, he has been artistic director and in 2007 he obtained the title of “Honored Artist of the Russian Federation”.

In 2011, at the invitation of BG Tevlin, he began working as an assistant professor in the subject “Practice of work with the children's choir” at the Department of Modern Choral Performing Arts of the Moscow Conservatory.