Much of the oblique humor comes from the subtle but consistent disparity in the brothers' pronunciation of "Shmenge". Together they played polka for an aging eastern-European immigrant audience, who typically dined on cabbage rolls and coffee (provided by the fictional Mrs. Vilve Yachke). Many of their songs were cover versions of popular songs. The duo also performed original songs, such as "There's Rhythm In My Lederhosen," and "Mama Cook Me Cabbage Rolls," and "The Cabbage Rolls and Coffee Polka."
The brothers were immigrants from the fictional country of Leutonia. The country is modelled after Lithuania (Lit. "Lietuva") and celebrates its Christmas with a symbolic egg, a feast of "falutniks", and the "exchanging of the socks". The Happy Wanderers got their start in Leutonian vaudeville playing the gelkis (an instrument, like the glass harmonica, made of glass jars, but played by smashing rather than sliding a finger along the wet rim), and took up polka during wartime.
The Shmenge Brothers were the subjects of the 1984 mockumentary The Last Polka. To promote The Last Polka, on February 26, 1985 the Shmenges guested on Late Night with David Letterman where they performed "Cabbage Rolls and Coffee Polka", which included a sing-along with the audience. Their appearance in the 1986 Comic Relief release was their last performance.