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 singularity type which appears is in the form of blow up for a sequences of solutions.  The singularities which can form for such a sequences of solutions should not be arbitrary, but have conjecturally obeyed the so called Energy Identity.  I will discuss recent joint work with Daniele Valtorta which resolved this problem.