In this talk we start by explaining the isospectral problem on closed surfaces and the compactness result for isometry classes of metrics of Osgood, Phillips and Sarnark. Then, we will pass to non-compact surfaces that are hyperbolic outside a compact set; we consider finite genus and a finite number of cusps and funnels. We will describe the spectra of these surfaces and the give definition of resonances. We consider the isoresonant problem on these surfaces. I will mention two compactness results: One for surfaces only with cusps when one restricts to a conformal class and one for surfaces only with funnels. The compactness result for isoresonant surfaces with funnels is due to D. Borthwick and P. Perry.