Acceleration of low- and medium-β heavy ions by means of superconducting (SC) linear accelerators (linacs) was made possible by the development, during four decades, of a particular class of cavities characterized by low operation frequency, several different shapes and different electromagnetic modes of operation. Their performance, initially rather poor in operating accelerators, have steadily increased along with the technological progress and nowadays the gap with the high-β, elliptical cavities is close to be filled. Initially confined to a very small number of applications, this family of cavities evolved in many directions becoming one of the most widespread in linacs. Nowadays it is present in the majority of superconducting radio-frequency ion linac projects worldwide. An overview of low- and medium-β SC cavities for heavy ions, focused on their recent evolution and achievements, will be given.