In the 1960s, during the so-called Thaw, Voznesensky frequently travelled abroad: to France, Germany, Italy, the United States and other countries. The popularity of Voznesensky, Yevgeny Yevtushenko and Bella Akhmadulina was marked by their performances in front of the adoring crowds numbering in the thousands at stadiums, in concert halls and universities. In doing so, he served as "a sort of unofficial Kremlin cultural envoy", according to "The New York Times". His poetry reading skills were profound. He competed with Laurence Olivier and Paul Scofield at one show in London. He criticized the 1968 invasion of Czechoslovakia. While in the United States, he met Allen Ginsberg, Arthur Miller, and Marilyn Monroe.