Simplicial surfaces describe the incidence relations between vertices, edges and faces of triangulated 2-dimensional manifolds in a purely combinatorial way. By considering only the incidences of edges and faces, simplicial surfaces are closely related to cubic graphs. In this paper we investigate how properties of simplicial surfaces and cubic graphs can be transferred to each other. Furthermore, we study embeddings of cubic graphs on simplicial surfaces and how they are connected to strong graph embeddings. For instance, 3-connected cubic planar graphs are uniquely embeddable on simplicial spheres, which is a direct consequence of Whitney's embedding theorem. Moreover, 3-connected cubic planar graphs can also be embedded on simplicial surfaces of higher genus. We characterise the properties that a simplicial sphere must possess such that the cubic graph describing its edge-face incidence relation can be embedded on a simplicial surface of non-negative Euler characteristic.