  We derive the complete $(curvature)^2$ terms of effective D-brane actions,
for arbitrary ambient geometries and world-volume embeddings, at lowest order
(disk-level) in the string-loop expansion. These terms reproduce the
$o(\alpha'^2)$ corrections to string scattering amplitudes, and are consistent
with duality conjectures. In the particular case of the D3-brane with trivial
normal bundle, considerations of $SL(2,\mathbb{Z})$ invariance lead to a
complete sum of D-instanton corrections for both the parity-conserving and the
parity-violating parts of the effective action. These corrections are required
for the cancellation of the modular anomalies of massless modes, and are
consistent with the absence of chiral anomalies in the intersection domain of
pairs of D-branes. We also show that the parity-conserving part of the
non-perturbative R^2 action follows from a one-loop quantum calculation in the
six-dimensional world-volume of the M5-brane compactified on a two-torus.
