Conceptual graph formalism is a knowledge representation language in AI based on a graphical form of logic. Although logic is the basis of the conceptual graph theory, there is a strongly felt absence of a formal treatment of conceptual graphs as a logic programming language. In this paper, we develop the notion of a conceptual graph program as a kind of graph-based order-sorted logic program. First, we define the syntax of the conceptual graph program by specifying its major syntactic elements. Then, we develop a kind of model theoretic semantics and fixpoint semantics of the conceptual graph program. Finally, we show that the two types of semantics coincide for the conceptual graph programs.