  The properties of Dirac gamma matrices in a four-dimensional space-time with
the $(2,2)$ signature are studied. The basic spinors are classified, and the
existence of Majorana-Weyl spinors is noted. Supersymmetry in $2 + 2$
dimensions is discussed, and the existence of the {\it real chiral} scalar
supermultiplet is discovered. Supersymmetric {\it self-dual} Yang-Mills
theories and {\it self-dual} supergravity model in $2 + 2$ dimensions, that are
apparently relevant to integrable systems, are formulated for the first time.
