Baroclinic instability is that of flows in a rotating stratified fluid with a vertical velocity shear. The generation of large-scale vortical flows in the atmospheres of Earth and other planets is associated with this instability. The review presents modern theoretical approaches dealing with this instability. They include a description of baroclinic instability through the interaction of edge Rossby waves, the study of the problem of optimal perturbations, i.e., those characterized by the largest growth of energy or other functionals, and an analysis of the nonlinear dynamics of perturbations that relies on a low-mode approximation of the Galerkin method. Classical energy criteria for the stability of zonal flows obtained by the direct Lyapunov—Arnold method are also considered. The results presented may be of interest to specialists in the field of continuum mechanics and astrophysics.