The purpose of this tutorial is to give an introduction to finite element exterior calculus, targeted to an audience which is reasonably familiar with topics like elliptic partial differential equations, Sobolev spaces, and finite element methods. We will first give a brief review of some of the fundamental concepts of exterior calculus, such as interior and exterior products, pullbacks, the Hodge star operation, the exterior derivative, and Stokes' theorem. Then we will focus on some of the main building blocks of finite element exterior calculus. In particular, we will discuss piecewise polynomial spaces of differential forms, degress of freedom, and the construction of bounded cochain projections. In addition, an abstract theory of Hilbert complexes will be presented, and we will explain how this relates to to the stability theory for approximations of the Hodge Laplacian.