Complexes of graph homomorphisms, Hom-complexes for short, are gadgets introduced by Lovasz to study topological obstructions to the existence of graph colorings. These complexes came as a natural generalization of neighborhood complexes, which were so spectacularly used by Lovasz in resolving the Kneser conjecture. In these talks I will give motivation for introducing complexes assigned to graphs. Then I will define Hom-complexes, and give an overview of their properties and the existing results on this topic.