There is only one compact complex curve of genus , the Riemann sphere. In the XIXth century M. Noether conjectured a similar statement in dimension : is every compact complex surface without nonzero holomorphic differential forms a rational surface? The first counterexample to this conjecture was given by F. Enriques in 1896. Enriques’ example led to new questions and conjectures about the existence of surfaces without nonzero holomorphic differential forms with special properties. I will report on the history of these questions, and construct some of these surfaces.