Summary: The article introduces the notion of an extremal state of a system which consists of a vector space, a topology in the space, a set of cones and a state set. The state is a point in this state set such that each ray from this .point in a direction contained within a certain cone contains no points of the state set near the original point. Sufficient conditions for the existence of extremal states and extremality criteria are given. Possible applications of this notion are described, in particular, in linear and convex programming, in continuous transport problems, in :the theory of economic models of supply, etc.