  This paper deals with classifying the dynamics of {\it Topologically Anosov}
plane homeomorphisms. We prove that a Topologically Anosov homeomorphism
$f:\mathbb{R}^2 \to \mathbb{R}^2$ is conjugate to a homothety if it is the time
one map of a flow. We also obtain results for the cases when the nonwandering
set of $f$ reduces to a fixed point, or if there exists an open, connected,
simply connected proper subset $U$ such that $U \subset \mathrm{Int}(\overline
{f(U)})$, and such that $ \cup_{n\geq 0} f^n (U)= \mathbb{R}^2$. In the general
case, we prove a structure theorem for the $\alpha$-limits of orbits with empty
$\omega$-limit (or the $\omega$-limits of orbits with empty $\alpha$-limit),
and we show that any basin of attraction (or repulsion) must be unbounded.
