More generally, in mathematics, for any manifold with a conformal structure (which assigns an angle to intersections of differentiable curves), a **conformal mapping** is any homeomorphism which preserves the conformal structure. For example, the cartographic example of projecting a 2-sphere onto the plane augmented with a point at infinity is a conformal map.

In particular, in complex analysis, a **conformal map** is a function *f* : *U* `->` **C** (where *U* is an open subset of the complex numbers **C**) which maintains angles, and therefore the shape of small figures. A function *f* is conformal if and only if it is holomorphic and its derivative is everywhere non-zero.

An important statement about conformal maps is the Riemann mapping theorem.

A map of the extended complex plane onto itself (the word *onto* means *surjective*) is conformal iff it is a Möbius transformation.