A wide variety of problems in Science, Engineering, Environment, Biomedicine and other applications are typically modeled by means of deterministic mathematical representations of the underlying natural laws. Nowadays, there is a trend to incorporate uncertainty in these descriptions to account for lack of knowledge on relevant data and parameters, random fluctuations of operating conditions or just ignorance of the details of a proper model. The goal is to predict quantities of interest while quantifying uncertainty in such predictions. We will focus on Bayesian formulations, due to their versatility in this context, considering two kinds of practical implementations. On one side, we will examine Markov Chain Monte Carlo techniques adapted to the analysis of posterior probabilities. On the other, we will discuss low cost methods based on the linearization of posterior probabilities about maximum a posteriori estimates found by constrained optimization techniques, the so-called Laplace approximation. We will illustrate the performance of these methods on selected case studies: parameter identification in compartmental models for epidemics, shear elastography for medical imaging, holographic microscopy and geophysical studies.