One way to encourage an exchange of ideas between the two disciplines is to bring the string physicists and interested mathematicians together at a conference. One such conference was held at the Mathematical Sciences Research Institute in Berkeley, California in March 1986. The lectures dealt primarily with the Riemann surface aspect of strings. We held a conference in La Jolla during July 1986 to catalyze the exchange of ideas started at the Berkeley conference and to expand the directions for pursuit of solutions to the mathematical problems involved. It is our pleasure to present the proceedings of this conference in this publication.