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A fruitful approach for providing effective statistical descriptions of chaotic dynamical systems is to lift the dynamics to linear evolution operators on infinite dimensional Banach spaces. Spectral data of these operators, known as Koopman or transfer operators, yield insight into the long-term behaviour of the underlying system. In this talk I will give a short introduction into the spectral theory of transfer operators associated to (mostly) low-dimensional discrete chaotic dynamical systems, and present a complete description of eigenvalues of these operators for a certain special class of non-linear maps. In general, such explicit descriptions of eigendata are rarely available, and in various applications algorithms or numerical schemes to approximate these are required. I will present strong spectral convergence and consistency results for a popular numerical approximation scheme – extended dynamic mode decomposition (EDMD) – applied to examples of chaotic systems given by one-dimensional expanding maps.