We consider the problem of extending and avoiding partial edge colorings of hypercubes; that is, given a partial edge coloring φ of the d-dimensional hypercube Qd, we are interested in whether there is a proper d-edge coloring of Qd that agrees with the coloring φ on every edge that is colored under φ; or, similarly, if there is a proper d-edge coloring that disagrees with φ on every edge that is colored under φ. In particular, we prove that for any d≥ 1 , if φ is a partial d-edge coloring of Qd, then φ is avoidable if every color appears on at most d/8 edges and the coloring satisfies a relatively mild structural condition, or φ is proper and every color appears on at most d- 2 edges. We also show that φ is avoidable if d is divisible by 3 and every color class of φ is an induced matching. Moreover, for all 1 ≤ k≤ d, we characterize for which configurations consisting of a partial coloring φ of d- k edges and a partial coloring ψ of k edges, there is an extension of φ that avoids ψ.