  This article is based in part on lecture notes prepared for the summer school
"The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles" at the
Institute for Mathematical Sciences at the National University of Singapore in
July of 2014. The aim is to provide a brief introduction to algebraic stacks,
and then to give several constructions of the moduli stack of Higgs bundles on
algebraic curves. The first construction is via a "bootstrap" method from the
algebraic stack of vector bundles on an algebraic curve. This construction is
motivated in part by Nitsure's GIT construction of a projective moduli space of
semi-stable Higgs bundles, and we describe the relationship between Nitsure's
moduli space and the algebraic stacks constructed here. The third approach is
via deformation theory, where we directly construct the stack of Higgs bundles
using Artin's criterion.
