In mathematics, a **Godeaux surface** is one of the surfaces of general type introduced by Lucien Godeaux. Other surfaces constructed in a similar way with the same Hodge numbers are also sometimes called Godeaux surfaces. Surfaces with the same Hodge numbers (such as Barlow surfaces) are called **numerical Godeaux surfaces**.

The cyclic group of order 5 acts freely on the Fermat surface of points (*w : x : y : z*) in *P*^{3} satisfying *w*^{5} + *x*^{5} + *y*^{5} + *z*^{5} = 0 by mapping (*w* : *x* : *y* : *z*) to (*w:ρx:ρ*^{2}y:ρ^{3}z) where ρ is a fifth root of 1. The quotient by this action is the original **Godeaux surface**.

The fundamental group (of the original Godeaux surface) is cyclic of order 5.