As a consequence of the large top quark Yukawa coupling, supersymmetric unified theories with soft supersymmetry breaking terms generated at the Planck scale predict lepton flavour and CP violating processes with significant rates. The flavour violating parameters of the low energy theory are derived in both SU(5) and SO(10) theories, and are used to calculate the rate for mu --> e gamma. The sensitivity of the search for mu --> e gamma is compared with that for mu --> e conversion in atoms, tau --> mu gamma and the electric dipole moment of the electron. The experimental search for these processes is shown to provide a very significant test of supersymmetric unification, especially in SO(10) but also in SU(5).