How to find a conformal map of the unit disk on a given simply-connected domain
The geometry processing group at Carnegie Mellon University recently developed an algorithm called Boundary First Flattening that allows you to efficiently and interactively compute conformal parameterizations of triangle meshes. You can download the software here: https://geometrycollective.github.io/boundary-first-flattening/
It is incredibly powerful and easy to use. Unlike almost all previous algorithms for conformal parameterization it allows for significant control over the boundary shape.
You may want to look at Don Marshall's Zipper algorithm: https://sites.math.washington.edu/~marshall/zipper.html
Added in Edit by T. Banakh. This Zipper algorithm yields the following image of the conformal map of the unit disk to an oak leaf.
Many thanks to Prof. Donald E. Marshall for producing this image (which I post here with his permission).
These guys wrote the software: http://gsl.lab.asu.edu/doc/surfacecm.html. They are the Geometry Systems Lab at Arizona State University.