We begin this talk with an introduction to linear and nonlinear harmonic maps between Riemannian manifolds, with a first goal of understanding some basic examples and uses in geometry and analysis. Unlike linear harmonic maps, which are always smooth and well behaved, nonlinear harmonic maps may exhibit singularities of various sorts. One type of singularity which appears is in the form of discontinuous points of a fixed solution. Another which appears is in the form of blow up not for a single solution, but for a sequences of solutions. We will give examples and discuss how they relate to the general theory, attempting to end at an explanation of the regularity theory and energy identity of nonlinear harmonic maps. Work discussed is joint with Daniele Valtorta.