Explicit Riemann mappings Typical proofs of the Riemann mapping theorem are not terribly explicit (one maximizes a functional, or something equivalent, such as using Dirichlet's principle).
The theorem states that if $U$ is a simply connected open subset of the plane, then there is a biholomorphism between $U$ and the unit disk. I imagine, due to the wild generality of the result, no explicit construction can be expected in general. However, in many concrete cases, I would think a construction "by hand" should be possible; and in applications (to problems in engineering, for example), this would almost be a requirement.
Do you know of such a construction, or of a reference where these constructions are discussed? (The answer may of course only apply to certain families of open sets.) 
I know of a very nice reference: "Schwarz-Christoffel Mapping", by Tobin A. Driscoll and Lloyd N. Trefethen, Cambridge Monographs on Applied and Computational Mathematics (No. 8). The Schwarz-Christoffel Mappings explicitly give us biholomorphisms between the upper half plane and the interior of simple polygons. I am hoping for additional examples. 
 A: This is a very big question, and a lot of work has been done on making the Riemann mapping theorem more explicit.  I have several comments:


*

*Böttcher coordinates provide explicit Riemann maps for the Fatou component containing a superattracting fixed point for a rational map.  In particular, a Riemann map for the complement of the filled Julia set of a quadratic polynomial with connected Julia set can be computed fairly explicitly.  Similarly, the Riemann map for the complement of the Mandelbrot set is fairly explicitly computable.

*Thurston and others have done some beautiful work involving approximating arbitrary Riemann maps using circle packings.  See Circle Packing: A Mathematical Tale by Stephenson.

*To some extent, constructing a Riemann map is simply a matter of constructing a harmonic function on a given domain (as well as the associated harmonic conjugate), subject to certain boundary conditions.  The solution to such problems is a huge topic of research in the study of PDE's, although the connection with Riemann maps is rarely mentioned.
Edit: Incidentally, if I wanted to construct a Riemann map explicitly on a given domain $D$, I would use the following PDE's approach.  First, translate the domain so that it contains the origin.  Next, use a numerical method to construct a harmonic function $F$ satisfying
$$
F(z) \;=\; -\log |z|
$$
for all $z\in\partial D$, and let
$$
R(z) = |z|e^{F(z)}.
$$
Then $R(0) = 0$, $R|_{\partial D} \equiv 1$, and $\log R$ is harmonic, so $R$ is the  radial component (i.e. modulus) of a Riemann map on $D$. The angular component can now be determined by the fact that its level curves are perpendicular to the level curves of $R$, and have equal angular spacing near the origin.
