Abstarct: In this thesis, we show that if H and K are finite groups with no common direct factor and G = H × K, then the structure and order of AutG can be simply expressed in terms of AutH, AutK and the central homomorphism groups Hom(H, Z(K)) and Hom(K, Z(H)). In section 3, we first find the automorphism group of the direct product of n copies of an indecomposable non-abelian group. We describe the automorphism group as matrices with entries which are homomorphisms between the n direct factors. We then use this descxription with a generalization of a result by Bidwell, Curran, and McCaughan on Aut(H×K), where H and K have no common direct factor, to provide structure and order theorems for an arbitrary direct product.