The first part of this paper presents actions for Dirichlet p-branes embedded in a flat tendimensional space-time. The fields of the (p + 1) -dimensional world-volume theories are the I OD space-time coordinates X, a pair of Majorana-Weyl spinors θ1 and 02, and a U(1) gauge field Aμ. The N = 2A or 2B super-Poincaré group in ten dimensions is realized as a global symmetry. In addition the theories have local symmetries consisting of general coordinate invariance of the world-volume, a local fermionic symmetry (called "kappa"), and U(1) gauge invariance. A detailed proof of the kappa symmetry is given that applies to all cases (p = 0, 1, . . . , 9). The second part of the paper presents gauge-fixed versions of these theories. The fields of the IOD ( p = 9) gauge-fixed theory are a single Majorana-Weyl spinor A and the U(1) gauge field Aμ. This theory, whose action turns out to be surprisingly simple, is a supersymmetric extension of 10D Born-Infeld theory. It has two global supersymmetries: one represents an unbroken symmetry, and the second corresponds to a broken symmetry for which A is the Goldstone fermion. The gauge-fixed supersymmetric D-brane theories with p > 9 can be obtained from the 10D one by dimensional reduction.