Abstract: When studying the representation theory of coalgebras or corings, one typically looks at the category of comodules. Yet there is another (co)representation category which - at least from the categorical point of view - is equally natural to consider. This is the category of contramodules. Contramodules were considered at the beginning of the coalgebra theory, but then they were neglected or even largely forgotten. In this talk we describe what contramodules are and how the recent developments in understanding of comodules (of a coring) yield additional insight into the structure of contramodules. The original new results presented in this talk were obtained in collaboration with Gabriella Bohm and Robert Wisbauer.