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.