  The first part of this paper presents actions for Dirichlet p-branes embedded
in a flat 10-dimensional space-time. The fields of the (p+1)-dimensional
world-volume theories are the 10d space-time coordinates $ X^m$, a pair of
Majorana-Weyl spinors $\theta_1$ and $\theta_2$, and a U(1) gauge field
$A_{\mu}$. The N = 2A or 2B super-Poincare 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 10d (p = 9) gauge-fixed theory are a single
Majorana-Weyl spinor $\lambda$ and the U(1) gauge field $A_{\mu}$. 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
$\lambda$ is the Goldstone fermion. The gauge-fixed supersymmetric D-brane
theories with $p<9$ can be obtained from the 10d one by dimensional reduction.
