Theoretical research will be conducted on quantum phase transitions, that is, phemonena in which the ground state of a many-body system changes qualitatively as some control parameter is varied at zero temperature. These include transitions from metal to insulator as a function of interaction strength, randomness or both, changes in magnetic order with hole density, representation of the spins, or microscopic couplings, and the transition from metal to superconductor with doping. All these problems have the common feature that there is generally no obvious small parameter to allow a clean theoretical calculation. This will be handled in a variety of ways: by working with low dimensional analogs amenable to exact solution, by embedding the given problem in a larger family (with bigger spin, bigger group, etc.) where a small parameter exists, and by the application of the renormalization group which permits one to deal with the problem in stages instead of all at once. %%% This work will utilize techniques developed for high energy physics to investigate phase transitions in condensed matter systems where the properties of matter change from one form to another, such as occurs when an insulating material changes to one which has no resistance to the flow of electric charge. Usually these phase transitions are studied as a function of temperature. However, for this work the calculations will be done at zero temperature where quantum effects dominate. The effects to be studied are basic to many modern problems in condensed matter physics and materials science.