A bielliptic surface over the complex numbers is a quotient of a product of elliptic curves by a finite group acting by a combination of translations and automorphisms of the elliptic curves. The study of these surfaces over number fields has played an important role in our understanding of rational points on algebraic varieties. I will review this history and then describe recent computations with Magma showing that Skorobogatov's famous bielliptic surface does indeed have a zero-cycle of degree 1, as predicted by a conjecture of Colliot-Thelene.