We generalise group algebras to other algebraic objects with bounded Hilbert space representation theory; the generalised group algebras are called ‘host’ algebras. The main property of a host algebra is that its representation theory should be isomorphic (in the sense of the Gelfand–Raikov theorem) to a specified subset of representations of the algebraic object. Here we obtain both existence and uniqueness theorems for host algebras as well as general structure theorems for host algebras. Abstractly, this solves the question of when a set of Hilbert space representations is isomorphic to the representation theory of a C*-algebra. To make contact with harmonic analysis, we consider general convolution algebras associated to representation sets, and consider conditions for a convolution algebra to be a host algebra.