We present a large set of new self-dual N = 1 SUSY gauge theories. Exa
mples include SU(N) theories with tensors and SO(N) theories with spin
ors. Using these dualities as starting points, new nontrivial duals ca
n be derived by breaking the gauge group through the Higgs mechanism o
r by integrating out matter. General lessons that can be learned from
these duals are that ''accidental'' infrared symmetries play an import
ant role in duality, many theories have more than one ''dual,'' and th
ere seems to be no simple organizing pattern which relates duals of th
eories with different number of flavors.