The Riemann Mapping Theorem is a fundamental result in classical complex analysis in one variable: If $\Omega\subset \mathbb C$ is a simply connected domain, $\Omega\neq \mathbb C$, then there is a biholomorphic map $F\colon \Omega\to\mathbb D:=\{|z|<1\}$. One of the first things we teach students in several complex variables is that the analogous fails miserably for domains in $\mathbb C^n$ for $n\geq 2$, as was already discovered by Poincaré; There is no biholomorphic map from the bidisk $\mathbb D^2:=\{(z_1,z_2)\colon |z_1|<1, |z_2|<1\}$ to the unit ball $\mathbb B^2=\{|z_1|^2+|z_2|^2<1\}$. There are clearly no topological obstructions to the existence, which is essentially the only obstruction to a Riemann map in one variable (but what about $\Omega\neq \mathbb C$?). As a first reaction, one might then give up and exclaim "if this example doesn't work, there is no hope for a reasonable Riemann Mapping Theorem in higher dimensions". Well, I intend to convince the audience that one would be wrong, and one would then miss an extremely rich theory that blends real and complex geometry, partial differential equations, and, of course, real and complex analysis.