Rainbow were one of the biggest bands among rock fans in Bulgaria. Actually, I was kind of surprised when I read your post and thought it was a joke - "there was a band called Rainbow"???
The band was created by Ritchie Blackmore after he left Deep Purple in 1974, before or during the recordings of the "Burn" album. He obviously wasn't happy with his role in the band (coincidently or not, "Burn" is considered the last really successful Deep Purple recording). So Ritchie Blackmore created his own band, and struck gold by getting Ronnie James Dio sing lead vocals (Dio was recommended to him by a friend who said something along the lines of "I have a singer for you who can stop a train with his voice").
They mostly sucked after 1978 (about the time when Dio left; he joined Black Sabbath in 1979 and recorded several albums with them before moving on to a solo career) and Ritchie Blackmore constantly changed the line up, but their early albums were great. They have some famous live recordings, especially "On Stage"  - I'd highlight their rendition of Deep Purple's "Mistreated", with very powerful vocals by Dio.
They also had a huge hit in 1980 with "I Surrender", which still gets a lot of radio play on retro radios (at least around here).