Intentions are an important concept in Artificial Intelligence and Cognitive Science. We present a formal theory of intentions and beliefs based on Discourse Representation Theory that captures many of their important logical properties. Unlike possible worlds approaches, this theory does not assume that agents are perfect reasoners, and gives a realistic view of their internal architecture; unlike most representational approaches, it has an objective semantics, and does not rely on an ad hoc labeling of the internal states of agents. We (...) describe a minimal logic for intentions and beliefs that is sound and complete relative to our semantics. We discuss several additional axioms, and the constraints on the models that validate them. (shrink)