Abstract: Given a geometric or algebraic structure, a moduli space can be used to study its automorphisms. In this talk, we consider two instances of such moduli spaces. In the first case, we examine automorphisms of right-angled Artin groups (RAAGs), a natural class of groups extending both free and free abelian groups. With Charney and Vogtmann, we construct a contractible, finite dimensional moduli space of locally CAT(0) complexes on which the automorphism group of a RAAG acts with finite stabilizers. In the second case, we study the embedding space of a link in R^3. In joint work with Boyd, we calculate the homotopy groups of these embedding spaces, and provide the first example computing the full homotopy type for a nontrivial link.