In this talk I first discuss the existence of eigenfunctions for the transfer operator of the Farey map. The main problem is that these operators are not quasi-compact on spaces of C^k functions for any k. However, on a suitable space of holomorphic functions, it is possible to have a characterization of the eigenfunctions. Then in the second part I will show that these eigenfunctions are related to Maass cusp and non-cusp forms of the modular surface, that is eigenfunctions of the hyperbolic Laplacian on the hyperbolic half-plane quotiented by the action of full modular group PSL(2,Z). This is joint work with Stefano Isola.