Abstract: I will start by reviewing the modern Bayesian approach to model selection, highlighting both its conceptual simplicity and the difficulties associated with its practical implementation. This will be illustrated in the context of the canonical problem of variable selection in the context of a Normal multiple regression model. The last part of the talk will be devoted to presenting new developments in the area, specifically on mixtures of $g$-priors, associated computational strategies, and various consistency results. This part of the talk draws from joint work with Berger, Clyde, Feng, and Molina.