The world and European football governing bodies said they would be banned "until further notice".
It means the Russian men's team will not play their World Cup play-off matches next month and the women's team have been banned from this summer's Euro 2022 competition.
Spartak Moscow have also been kicked out of the Europa League and their last-16 opponents RB Leipzig will advance to the quarter-finals.
Uefa has also ended its sponsorship with Russian energy giant Gazprom.
