In this lecture, Michael Crandall provides an excellent expository introduction to the theory of viscosity solutions of partial differential equations. The theory, which applies to scalar fully nonlinear PDEs of the form $F(x, u, Du, D^2u)=0$, has yielded very general existence and uniqueness theorems. The theory provides an elegant and efficient way to approach cases in which $F$ is a first order Hamilton-Jacobi equation, as well as cases in which $F=0$ is a fully nonlinear uniformly elliptic second order equation. The theory also admits the possibility of solutions $u$ which are nowhere differentiable. Accessible to a general mathematical audience, the lecture provides a good balance of theory and examples. This topic provides an introduction into the study of PDEs without an overwhelming dose of technical machinery. Crandall makes judicious choices about which technical details to include and which to leave out, making the lecture suitable for beginning graduate students or advanced undergraduates.