Galileon theories are of considerable interest since they allow for stable violations of the null energy condition. Since such violations could have occurred during a high-energy regime in the history of our Universe, we are motivated to study supersymmetric extensions of these theories. This is carried out in this paper, where we construct generic classes of N=1 supersymmetric Galileon Lagrangians. They are shown to admit nonequivalent stress-energy tensors and, hence, vacua manifesting differing conditions for violating the null energy condition. The temporal and spatial fluctuations of all component fields of the supermultiplet are analyzed and shown to be stable on a large number of such backgrounds. In the process, we uncover a surprising connection between conformal Galileon and ghost-condensate theories, allowing for a deeper understanding of both types of theories.