  We review the specific problems that arise when studying instantons on a
torus. We discuss how the Nahm transformation shows that no exact charge one
instanton on T**4 can exist. However, taking one of the directions (the time)
to infinity, it can be shown that vacuum to vacuum tunnelling solutions exist.
A precise description of the moduli space for T**3xR, studied numerically using
lattice techniques, remains an interesting open problem. New is an explicit
application of the Nahm transformation to (anti-)selfdual constant curvature
solutions on T**4 and a discussion of its properties relevant to instantons on
T**3xR.
