Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.

The following are some of the most important topics in geometric function theory:

A conformal map is a function which preserves angles locally. In the most common case the function has a domain and range in the complex plane.

More formally, a map,

is called conformal (or angle-preserving) at a point ${\displaystyle u_{0}}$ if it preserves oriented angles between curves through ${\displaystyle u_{0}}$ with respect to their orientation (i.e., not just the magnitude of the angle). Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature.

In mathematical complex analysis, a quasiconformal mapping, introduced by Grötzsch (1928) and named by Ahlfors (1935), is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity.

Intuitively, let f : D → D′ be an orientation-preserving homeomorphism between open sets in the plane. If f is continuously differentiable, then it is K-quasiconformal if the derivative of f at every point maps circles to ellipses with eccentricity bounded by K.

If K is 0, then the function is conformal.