  In this paper, we give an expository account of the geometric properties of
the moduli stack of $G$-bundles. For $G$ an algebraic group over a base field
and $X \to S$ a flat, finitely presented, projective morphism of schemes, we
give a complete proof that the moduli stack $Bun_G$ is an algebraic stack
locally of finite presentation over $S$ with schematic, affine diagonal. In the
process, we prove some properties of $BG$ and Hom stacks. We then define a
level structure on $Bun_G$ to provide alternative presentations of
quasi-compact open substacks. Finally, we prove that $Bun_G$ is smooth over $S$
if $G$ is smooth and $X \to S$ is a relative curve.
