  We construct the half-supersymmetric instanton solutions that are
electric-magnetically dual to the recently discussed half-supersymmetric
Q7-branes. We call these instantons `Q-instantons'. Whereas the D-instanton is
most conveniently described using the RR axion \chi and the dilaton \phi, the
Q-instanton is most conveniently described using a different set of fields
\chi' and T, where \chi' is an axionic scalar. The real part of the Q-instanton
on-shell action is a function of T and the imaginary part is linear in \chi'.
Discrete shifts of the axion \chi' correspond to PSL(2,Z) transformations that
are of finite order. These are e.g. pure S-duality transformations relating
weak and strongly coupled regimes. We argue that near each orbifold point of
the quantum axion-dilaton moduli space PSL(2,Z)\PSL(2,R)/SO(2) the higher order
R^4 terms in the string effective action contain contributions from an infinite
sum of single multiply-charged instantons with the Q-instantons corresponding
to the orbifold points \tau=i,\rho where \tau is the complex axion-dilaton
field.
