We present a categorical theory of 'well-behaved' operational semantics which aims at complementing the established theory of domains and denotational semantics to form a coherent whole. It is shown that, if the operational rules of a programming language can be modelled as a natural transformation of a suitable general form, depending on functorial notions of syntax and behaviour, then one gets the following for free: an operational model satisfying the rules and a canonical, internally fully abstract denotational model which satisfies the operational rules. The theory is based on distributive laws and bialgebras; it specialises to the known classes of well-behaved rules for structural operational semantics, such as GSOS.