Lots of graphs having a symmetry property can be described as cover-ings of simpler graphs. In this manuscript, we examine several enumeration problems for various types of nonisomorphic graph coverings of a graph and some of their applications to a group theory or to a surface theory. This manuscript is organized as follows. In chapter 1, we introduce basic concepts. In chapter 2, by using covering graph construction, we count the positive isomorphism classes of cycle permutation graphs, which is equal to the number of double cosets of the dihedral group Dn in the symmetric group Sn on n elements. In chapter 3, we count nonisomorphic (connected) coverings of a graph and, as its application, we have another recursive formula for the number of conjugacy classes of subgroups of given index of a finitely generated free group. In chapter 4, we count nonisomorphic regular coverings of a graph whose covering transformation groups are abelian and, as its application, we count subgroups of given index of free abelian groups. The same work is done in chapter 5 for regular coverings having dihedral voltage groups. In chapter 6, we discuss a general counting formula for regular coverings having any finite voltage group. In chapter 7, after discussing a combinatorial proof of Hurwitz theorem for surface branched coverings, we consider the number of subgroups of surface groups. Finally, in chapter 8, we discuss a distribution of branched surface coverings of surfaces and some related topological properties including a generalization of the classical Alexander theorem.