We present an approach to the interpretation of non-sentential utterances like B's utterance in the following mini-dialogue:

A: ``Who came to the party?''
B: ``Peter.´´

Such utterances pose several puzzles: they convey `sentence-type' messages (propositions, questions or request) while being of non-sentential form; and they are constrained both semantically and syntactically by the context. We address these puzzles in our approach which is compositional, since we provide a formal semantics for such fragments independent of their context, and constraint-based because resolution is based on collecting contextual constraints.