This is an introductory course on Bayesian methods for computational inverse problems, including parameter and state estimation problems. In the Bayesian framework, the unknowns are modeled as random variables, where the randomness reflects the lack of certainty about the value, and the observations are augmented by any possible prior information about the unknown. The course covers the basic Bayesian techniques like defining priors and likelihoods, and computational techniques for solving for the posterior densities. Sampling methods including Markov chain Monte Carlo (MCMC) and Bayesian filtering techniques are included.