  We classify all smooth flat Riemannian metrics on the two-dimensional plane.
In the complete case, it is well-known that these metrics are isometric to the
Euclidean metric. In the incomplete case, there is an abundance of
naturally-arising, non-isometric metrics that are relevant and useful.
Remarkably, the study and classification of all flat Riemannian metrics on the
plane -- as a subject -- is new to the literature. Much of our research focuses
on conformal metrics of the form $e^{2\varphi}g_0$, where $\varphi :
\mathbb{R}^2 \rightarrow \mathbb{R)$ is a harmonic function and $g_0$ is the
standard Euclidean metric on $\mathbb{R}^2$. We find that all such metrics,
which we call "harmonic", arise from Riemann surfaces.
