This paper proposes a general query language for conceptual graphs. First, we introduce kernel query graphs. A kernel query graph can be used to express an “or” between two subgraphs, as well as an “option” on an optional sub-graph. Second, we propose a way to express four kinds of queries (ask, select, describe and construct) using kernel query graphs. Third, the answers of queries are computed by an operation based on graph homomorphism.