We construct N = 1 supergravity extensions of scalar field theories with higher-derivative kinetic terms. Special attention is paid to the auxiliary fields, whose elimination leads not only to corrections to the kinetic terms, but to new expressions for the potential energy as well. For example, a potential energy can be generated even in the absence of a superpotential. Our formalism allows one to write a supergravity extension of any higher-derivative scalar field theory and therefore has applications to both particle physics and cosmological model building. As an illustration, we couple the higher-derivative Dirac-Born-Infeld action describing a 3-brane in six dimensions to N = 1 supergravity. This displays a number of new features including the fact that in the regime where the higher-derivative kinetic terms become important, the potential tends to be everywhere negative.