Zveri was formed in mid-2002 by Roman Bilyk, the lead singer of the band, better known as Roma Zver'. The band was named after his nickname. He was soon joined by Maksim Leonov on guitars, Kostya Labezkiy on the bass, Kirill Antonenko on keyboards and Misha Kraev on the drums and the group began work on their debut album Golod (Hunger), released in February 2003. This album contained many of the group's best-known songs, such as Dozhdi-Pistolety (Rain-Guns), Dlya Tebya (For You) and Prosto Takaya Sil'naya Lyubov' (Simply such a strong love). In March 2004, Zveri released their second album, entitled Rayony-Kvartaly (Regions-districts). It sold well and was accompanied by a tour of the country to promote it. Vsyo, chto kasaetsya (All, that is about) was released as a single from it and standout tracks include Yuzhnaya Noch' (Southern Night) and Napitki Pokrepche (stronger drinks). In 2005 the group released a remix album for songs from Golod and Rayony, Kvartaly. In 2004, Zveri was awarded Muz-TV prize as Best rock act, and since then received it six times in a row, and again in 2012. MTV Russia also awarded Zveri in 2005.