In mathematics, a two-dimensional surface is a geometrical shape that resembles a deformed plane. The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R³ such as spheres. The exact definition of a surface may depend on the context. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not.

The simplest examples of two-dimensional surfaces can be obtained by gluing the sides of a square according to different rules. Above are surfaces called a cylinder, a torus, a Klein bottle and a projective plane, respectively.