  We revisit two longstanding puzzles in supersymmetric gauge theories. The
first concerns the question of the holomorphy of the coupling, and related to
this the possible definition of an exact (NSVZ) beta function. The second
concerns instantons in pure gluodynamics, which appear to give sensible, exact
results for certain correlation functions, which nonetheless differ from those
obtained using systematic weak coupling expansions. For the first question, we
extend an earlier proposal of Arkani-Hamed and Murayama, showing that if their
regulated action is written suitably, the holomorphy of the couplings is
manifest, and it is easy to determine the renormalization scheme for which the
NSVZ formula holds. This scheme, however, is seen to be one of an infinite
class of schemes, each leading to an exact beta function; the NSVZ scheme,
while simple, is not selected by any compelling physical consideration. For the
second question, we explain why the instanton computation in the pure
supersymmetric gauge theory is not reliable, even at short distances. The
semiclassical expansion about the instanton is purely formal; if infrared
divergences appear, they spoil arguments based on holomorphy. We demonstrate
that infrared divergences do not occur in the perturbation expansion about the
instanton, but explain that there is no reason to think this captures all
contributions from the sector with unit topological charge. That one expects
additional contributions is illustrated by dilute gas corrections. These are
infrared divergent, and so difficult to define, but if non-zero give order one,
holomorphic, corrections to the leading result. Exploiting an earlier analysis
of Davies et al, we demonstrate that in the theory compactified on a circle of
radius beta, due to infrared effects, finite contributions indeed arise which
are not visible in the formal limit that beta goes to infinity.
