For graphs G and H , an H -coloring of G, or homomorphism from G to H , is an edge-preserving map from the vertices of G to the vertices of H . H -colorings generalize such graph theory notions as proper colorings and independent sets. In this dissertation, we consider four questions involving H -colorings of graphs. Recently, Galvin [27] showed that the maximum number of independent sets in an n vertex minimum degree δ graph occurs (for sufficiently large n) when G = Kδ,n−δ. First, we show this result holds for level sets: for all triples (n, δ, t) with δ ≤ 3 and t ≥ 3, no n-vertex graph with minimum degree δ admits more independent sets of size t than Kδ,n−δ, and we obtain the same conclusion for δ > 3 and t ≥ 2δ + 1. Second, we begin the project of generalizing Galvin?s result to arbitrary H . Writing hom(G, H ) for the number of H -colorings of G, we show that for δ = 1 and δ = 2 and fixed H , hom(G, H ) ≤ max{hom(Kδ+1, H ) n/δ+1 , hom(Kδ,δ, H ) 2δ , hom(Kδ,n−δ, H )} for any n vertex minimum degree δ graph G (for sufficiently large n). For δ ≥ 3 (and sufficiently large n), we provide a class of H for which hom(G, H ) ≤ hom(Kδ,n−δ, H ) for any G in this family. Third, for a given H , k ∈ V (H ), and regular bipartite G, we consider the pro- portion of vertices of G that get mapped to k in a uniformly chosen H -coloring of G. We find numbers 0 ≤ a−(k) ≤ a+(k) ≤ 1 with the property that for all such G, with high probability the proportion is between a−(k) and a+(k), and we give examples where these extremes are achieved. Fourth, we study the set of H -colorings of the even discrete torus Zdm. For any H and fixed m, we show that the space of H -colorings of Zdm may be partitioned into a subset of negligible size (as d grows) and a collection of subsets indexed by certain pairs (A, B) ∈ V (H)2, with each H -coloring in the subset indexed by (A, B) having almost all vertices in one partition class mapped to A and almost all vertices in the other partition class mapped to B.