**1. Introduction**

Inverse problems arise in many scientific and engineering applications. In fact, almost always we want to infer on quantities, variables, distributions and functions which are not directly observable. Inferring on a hidden variable *f* via the observation of another variable *g* is the main objective in many scientific area [1–3].

Classically, the Bayesian methods have been developed for direct observation of a quantity, its parametric modeling and the estimation of the parameters of the model. Description and development of the Bayesian inference for the case of inverse problems is the main objective of this chapter. The chapter is organized as follows: First a few inverse problems are mentioned, mainly in two categories: those described by Ordinary Differential Equations (ODE) and Partial Differential Equations (PDE)'s and those described with integral equations. Then, we will see that two main problems arise: parameter estimation and inversion. First the Bayesian parameter estimation is described and then the Bayesian inversion.
