The goal of these lecture notes is to review the problem of free energy minimization as a unified framework underlying the definition of maximum entropy modelling, generalized Bayesian inference, learning with latent variables, statistical learning analysis of generalization,and local optimization. Free energy minimization is first introduced, here and historically, as a thermodynamic principle. Then, it is described mathematically in the context of Fenchel duality. Finally, the mentioned applications to modelling, inference, learning, and optimization are covered starting from basic principles.