Dmitry Medvedev praised the 50-year-old lead singer of U2, six years Medvedev's senior, for writing music "that unites generations on many continents."
The president, a well-known fan of Deep Purple, chatted in English with Bono, whose band plays its first-ever concert in Russia on Wednesday.
Medvedev, in jeans and a shirt, and Bono, in dark glasses and trademark stubble, sipped tea at a table on a verandah.
Bono, who has worked to fight poverty, AIDS and other diseases in Africa, told Medvedev that small donations could eliminate the transfer of HIV from mother to child by 2015.
Medvedev said Bono was "doing important things."