In several applications the solutions of combinatorial optimization problems (COP) are required to satisfy an additional cardinality constraint, that is to contain a fixed number of elements. So far the family of (COP) with cardinality constraints has been little investigated. The present work tackles a new problem of this class: the k-cardinality minimum cut problem (k-card cut). For a number of variants of this problem we show complexity results in the most significant graph classes. Moreover, we develop several heuristic algorithms for the k-card cut problem for complete, complete bipartite, and general graphs. Lower bounds are obtained through an SDP formulation, and used to show the quality of the heuristics. Finally, we present a randomized SDP heuristic and numerical results.