The theory of scalar first-order fully nonlinear partial differential equations has recently enjoyed a strong development. One major step was a proof by M. Crandall and P.-L. Lions of the uniqueness of certain generalized solutions-called 'viscosity solutions'- of problems involving such equations with the scope to accommodate applications to, for example, differential games. Following this event there has been a continuous stream of work concerning the existence, approximation and representation of viscosity solutions of Hamilton-Jacobi equations as well as interaction of the theory of viscosity solutions and areas of application (primarily control theory and differential games), and refined uniqueness results. This survey paper, which corresponds to an invited address by the first author at an international symposium on differential equations held in March 1983 at the University of Alabama-Birmingham, introduces the relevant concepts and describes the major results up to, roughly, July 1983. (Author).