  In this paper we give both an historical and technical overview of the theory
of Harnack inequalities for nonlinear parabolic equations in divergence form.
We start reviewing the elliptic case with some of its variants and geometrical
consequences. The linear parabolic Harnack inequality of Moser is discussed
extensively, together with its link to two-sided kernel estimates and to the
Li-Yau differential Harnack inequality. Then we overview the more recent
developements of the theory for nonlinear degenerate/singular equations,
highlighting the differences with the quadratic case and introducing the
so-called intrinsic Harnack inequalities. Finally, we provide complete proofs
of the Harnack inequalities in some paramount case to introduce the reader to
the expansion of positivity method.
