Abstract: Numerical algebraic geometry (NAG) is a tool set for approximating the solutions to systems of polynomial equations and manipulating the resulting solution sets. Since polynomial equations arise frequently in science and engineering, techniques from NAG have been applied to various problems in these areas. This introduction to NAG will focus on the fundamental component of most NAG algorithms, numerical homotopy continuation. From this foundation, higher level algorithms will be discussed in the context of applications. Specific applications include finding exceptional mechanisms (kinematics), finding unit distance embeddings of graphs, and computing topological neighborhoods of spline curves.