We present a sound and complete semantics for the monitor concepts of C. A. R. Hoare. First a method for specification of monitors, introduced by O.-J. Dahl, is reviewed. This method is based on the relation between the historic sequence of monitor procedure calls and the historic sequence of monitor procedure exits. Based on such specifications and our new monitor semantics we present a method by which it is possible to prove that a concrete monitor is an implementation of an abstract one. In the last part of the paper an axiomatic semantics for systems of concurrent processes and monitors is introduced. The method supports verification by separation of concerns: Properties of the communication to and from each process are proven in isolation by a usual Hoare style axiomatic semantics, while abstract monitors are also specified in isolation by the method reviewed in the first part of the paper. These properties of the components of the system are then used in a new proof rule to conclude properties of the complete system.