The link of a vertex v of a graph G is the subgraph induced by the set of vertices of G adjacent to v. If all the links of G are isomorphic to a given graph H, then G is called locally H, or locally homogeneous. We deal with the problem of characterization of all connected locally graphs forH fixed. It turns out that the problem is closely related to the theory of covering spaces. This fundamental observation is generalized and developed in Section 1 of the paper. Theoretical results from Section 1 are used for deriving of some new results on locally homogeneous graphs and reproving of some old ones. For instance, Theorem 2.7 establishes that if a graph H has a limited number of edges then either there is no locally H graph, or there are infinitely many finite connected locally H graphs.