“Genus” has number of distinct but compatible definitions.

In topology, if is an orientable surface, its genus is the number of “handles” it has. More precisely, from the classification of surfaces, we know that any orientable surface is a sphere, or the connected sum of tori. We say the sphere has genus 0, and that the connected sum of tori has genus (alternatively, genus is additive with respect to connected sum, and the genus of a torus is 1). Also, where is the Euler characteristic of .

In algebraic geometry, the genus of a smooth projective curve over a field is the dimension over of the vector space of global regular differentials on . Recall that a smooth complex curve is also a Riemann surface, and hence topologically a surface. In this case, the two definitions of genus coincide.