Why is the Riemann mapping theorem important? The Riemann mapping theorem is as follows: Let $U \neq \mathbb{C}$ be a simply connected domain and $w_{1}, w_{2} \in U$ any points. Then, there exists a unique conformal mapping $f: \mathbb{D} \rightarrow U$ such that $f^{-1}(w_{1}) = 0$ and $f^{-1}(w_{2}) > 0$ (where $\mathbb{D}$ is the unit disk).
I would like to know the reason why the Riemann mapping theorem is so important. In particular I am curious to know if it is of interest to calculate the aforementioned function $f$ and if there is a technique to do it.
 A: For the importance of the Riemann mapping theorem, see Wikipedia.
The Riemann mapping theorem can be generalized to the biholomorphic classification of Riemann surfaces: these are essentially the Riemann sphere, the whole complex plane, and the open unit disc. This classification is known as the uniformization theorem.
For methods to compute $f$ that are useful in applications, see the Schwarz–Christoffel mapping.
A: According to my view, this theorem is highly non-trivial due to the following reasons:
the theorem itself is important. From the planar topology, we know that there are objects which is simply connected with complicated boundaries. There are no obvious homeomorphism between each other. While Riemann mapping theorem speaks that they are not only homeomorphic, but also bi-holomorphic. This is really amazing fact.  
A: Another way of stating the Riemann mapping theorem (RMT) is that for a simply connected region $\Omega\subset\mathbb{C}$, we have a unique analytic bijection $f:\Omega\to\mathbb{D}$ where $\mathbb{D}$ is the unit disk. Some important applications, corollaries and uses of the Riemann mapping theorem are as follows:


*

*The uniformisation theorem states that if $\Omega$ is a simply connected open subset of the Riemann surface, then $\Omega$ is biholomorphic to either the Riemann sphere, the complex plane, or the unit disk. It is a generalisation of the RMT to Riemann surfaces.

*Take two simply connected subsets of the Riemann sphere. They can be conformally mapped to each other via the RMT. 

*Take hard/complicated, simply connected open sets (that may have annoying constraints on the boundary). They can be mapped into the unit disk via an angle-preserving/conformal manner by the RMT. 

*Introduction of the measurable Riemann mapping theorem which is used in the study of the dynamical properties of polynomials, and to perform quasiconformal deformations. 

