  After reviewing how Albert Einstein's general relativity (GR) can be viewed
as a gauge theory of the Poincar\'e algebra, we show how \'Elie Cartan's
geometric formulation of Newtonian gravity (Newton-Cartan gravity) can be
viewed as a gauge theory of the Bargmann algebra following the construction of
[R. Andringa, E. Bergshoeff, S. Panda, and M. de Roo, "Newtonian Gravity and
the Bargmann Algebra," Class. Quant. Grav. 28 (2011) 105011, arXiv:1011.1145
[hep-th]]. In doing so, we will touch on the following auxiliary topics: the
extension of Yang-Mills gauge theory to a more generic formalism of gauge
theory, the fiber bundle picture of gauge theory along with the soldering
procedure necessary to complete the gravity as a gauge theory picture, the
vielbein formalism of GR, Lie algebra procedures such as central extensions and
\.In\"on\"u-Wigner contractions, and the hallmarks of Newtonian gravity which
differentiate it from GR. The objective of the present work is to pedagogically
fill in the gaps between the above citation and an undergraduate physics and
mathematics education. Working knowledge of GR and some familiarity with
classical field theory and Lie algebras is assumed.
