  We study supersymmetry and self-duality in a four-dimensional space-time with
the signature (2,2), that we call the Atiyah-Ward space-time. Dirac matrices
and spinors, in particular Majorana-Weyl spinors, are investigated in detail.
We formulate $ N\ge 1 $ supersymmetric self-dual Yang-Mills theories and
self-dual supergravities. An N=1 ``self-dual'' tensor multiplet is constructed
and a possible ten-dimensional theory that gives rise to the four-dimensional
self-dual supersymmetric theories is found. Instanton solutions are given as
the zero modes in the N=2 self-dual Yang-Mills theory. The N=2 superstrings are
conjectured to have no possible counter-terms at quantum level to all orders.
These self-dual supersymmetric theories are to generate exactly soluble
supersymmetric systems in lower dimensions.
