A face-transitive surface is a triangulated 2-dimensional manifold whose automorphism group acts transitively on its set of triangles. In this paper, we investigate this class of highly symmetric surface triangulations. We identify seven types of such face-transitive surfaces, splitting up further into a total of thirteen sub-types, distinguished by how their automorphism groups act on them. We use these theoretical results to compute a census of face-transitive surfaces with up to 1280 faces by constructing suitable cycle double covers of cubic node-transitive graphs.