This RUI award is for support of studies in quantum field theory. The objectives of the research are to apply nonlocal regularization methods that preserve appropriate symmetries to two classes of field theories. One class consists of discrete field theories known as lattice theories where one encounters difficulties in maintaining a right-left symmetry known as chiral symmetry. The other class consists of theories characterized as supersymmetric where one encounters difficulties in maintaining the underlying symmetry between two types of particles known as fermions and bosons. This research is directed to important problems at the frontier of elementary particle theory.