|Abstract:|| I will present different nonparametric methods for data distributed over complex spatial domains. First I will consider the problem of density estimation. Specifically I will propose a nonparametric penalized likelihood approach for data distributed over planar domains with complex geometries. The model formulation is based on a regularization with differential operators, and it is made computationally tractable by means of finite elements. In this setting, I will describe a permutation procedure for one and two samples hypothesis testing. Then I will consider hypothesis testing procedures in the case of spatial regression models with differential regularization. In particular, I will propose a test based on sign-flipping. I will present the performances of the proposed methods via simulation studies and application to real data.