According to the classification theorem of closed surfaces (2-manifold), any closed surface is homeomorphic to some member of one of these three families.

1. sphere

2. connected sum of g tori, g>=1

3. connected sum of k real projective planes for k>=1

1 and 2 are orientable and 3 is non-orientable.

For 1 and 2, a surface that is the connected sum of n tori is said to be of genus n, while a sphere is of genus 0 (Intuitively speaking, a genus number corresponds a number of handles in a closed orientable surface).

This genus number is also involved in Euler characteristic, which is topologically invariant. So we can say that we classify closed orientable 2-manifolds up to homeomorphism by using a genus.