They have produced many well-known hits in Denmark, like Vuffeli-vov, Hvalen Hvalborg, Står på en Alpetop, Stærk Tobak (to the music from the Phil Medley and Bert Russell song Twist and Shout), Familien kom til Kaffe, Sexchikane, We wanna be free, and many more.

They are the Danish band with the most albums sold in Denmark, with more than 6 million as of 2022. Notable characteristics were word plays and humour, but also often with serious backgrounds - like sexual harassment, climate change, or life.

Bundesen named "Den røde tråd" a comical but still autobiographical song about his development from working-class poverty to finding "your place in the sun."

The group was extended with the singer Michael Bundesen and keyboard player Jens Tage Nielsen [da], and under the name Shu-bi-dua released the single Fed Rock/Tynd Blues before the group released the single Change of the Guard/Highway of Living under the name Passport.

Shu-bi-dua was originally considered a one-off group, but when the debut single produced the hit Fed Rock, they chose to keep it.

One reason for their success was their often humorous translations of rock classics, made with great care and accuracy and often with a refined sense of wordplays in their Danish lyrics (largely untranslatable to English).

In 1988, Peter Andersen [da] replaced Paul Callaby as drummer, and shortly after that, Søren Jacobsen left the band again.

In the intervening years, they re-released their back catalogue on CD in 1990, and in 1991, released a Best Of album by the name Stærk tobak!!