In this paper we describe a new method for analyzing the Laplacian on asymptotically hyperbolic, or conformally compact, spaces, which was introduced in [Vasy 2010a]. This new method in particular constructs the analytic continuation of the resolvent for even metrics (in the sense of [Guillarmou 2005]), and gives high-energy estimates in strips. The key idea is an extension across the boundary for a problem obtained from the Laplacian shifted by the spectral parameter. The extended problem is nonelliptic—indeed, on the other side it is related to the Klein–Gordon equation on an asymptotically de Sitter space—but nonetheless it can be analyzed by methods of Fredholm theory. In [Vasy 2010a] these methods, with some additional ingredients, were used to analyze the wave equation on Kerr-de Sitter space-times; the present setting is described there as the simplest application of the tools introduced. The purpose of the present paper is to give a self-contained treatment of conformally compact spaces, without burdening the reader with the additional machinery required for the Kerr-de Sitter analysis.