In this talk, I will review the categorical approach to Information Geometry started in the 70's by Chentsov. Information geometry is a method of exploring the world of information by differential geometry, mainly Riemannian geometry. In this setting, the notion of a statistical model is the departure one and many properties of statistical inference can be interpreted as geometrical properties of the associated manifolds. In particular, a distinguished role in this theory is played by the Fisher-Rao metric tensor, which ubiquitously appears in estimation theory. Chentsov interpreted this metric tensor using a categorical approach: The Fisher-Rao metric tensor is the unique invariant tensor under a family of transformations forming the morphisms of a category. This approach to information theory was also extended to the quantum setting. In this case, however, the Riemannian metric tensors which are monotone with respect to completely positive trace-preserving maps are characterized by an operator-monotone function, and many different metric tensors have been employed to address different quantum problems. In the last part of the talk, I will present a different category, which is called the NCP category, where one can deal at the same time with classical and quantum systems. In this setting, one can consider a generalized version of a statistical model, which is provided by Lie categories embedded into the NCP one. As a first consequence, one can derive an analogous Cramer-Rao bound for estimators of these models in terms of a symmetric form on the algebroid associated with the Lie category.