  We address the problem of testing the dimensionality of classical and quantum
systems in a `black-box' scenario. We develop a general formalism for tackling
this problem. This allows us to derive lower bounds on the classical dimension
necessary to reproduce given measurement data. Furthermore, we generalise the
concept of quantum dimension witnesses to arbitrary quantum systems, allowing
one to place a lower bound on the Hilbert space dimension necessary to
reproduce certain data. Illustrating these ideas, we provide simple examples of
classical and quantum dimension witnesses.
