Topics in Supersymmetry and Supergravity.

Various aspects of supersymmetric theories are examined. We explain why supersymmetry might be an important ingredient for high energy theory and review some of the formalism that has proven useful in handling supersymmetric theories. We then apply this to the problem of finding the supersymmetric extension to the fourth order gravity lagrangian, which is known to lead to a renormalizable theory. We calculate the one loop effective action for locally supersymmetric theories and demonstrate how the fourth order terms, which render the theory non-renormalizable, are generated. We also show that, at least in a large class of locally supersymmetric grand unified theories, the quantum corrections do not upset the tree level gauge hierarchy. Finally, we address the important problem of experimental verification of supersymmetry. Photinos as a candidate for dark matter are considered. Pair annihilation into a heavy quark-antiquark bound state plus a monochromatic photon can produce potentially observable sharp peaks in the diffuse cosmic (gamma)-ray background.