This paper is a follow up of the author’s programme of characterizing Ramsey classes of structures by a combination of model theory and combinatorics. This relates the classification programme for countable homogeneous structures to the proof techniques of the structural Ramsey theory. Here we consider the classes of topological and metric spaces which recently were studied in the context of extremally amenable groups and of the Urysohn space. We show that Ramsey classes are essentially classes of finite objects only. While (...) for Ramsey classes of topological spaces we achieve a full characterization, for metric spaces this seems to be at present an intractable problem. (shrink)