  A detailed understanding of instanton effects for half-BPS couplings is
pursued in theories with 16 supersymmetries. In particular, we investigate the
duality between heterotic string on $T^4$ and type IIA on $K_3$ at the
$T^4/Z_2$ orbifold point, as well as their higher and lower dimensional
versions. We present a remarkably clean quantitative test of the duality at the
level of $F^4$ couplings, by completely matching a purely one-loop heterotic
amplitude to a purely tree-level type II result. The triality of SO(4,4) and
several other miracles are shown to be crucial for the duality to hold. Exact
non-perturbative new results for type I', F on $K_3$, M on $K_3$, and IIB on
$K_3$ are found, and the general form of D-instanton contributions in type IIA
or B on $T^4/Z_2$ is obtained. We also analyze the NS5-brane contributions in
type II on $K_3\times T^2$, and predict the value $\mu (N)=\sum_{d|N} (1/d^3)$
for the bulk contribution to the index of the NS5-brane world-volume theory on
$K_3 \times T^2$.
