Baroclinic instability is the instability of the flows of a stratified rotating fluid medium with a vertical shear of velocity. Large-scale vortex formation in the atmosphere of Earth and other planets is associated with the realization of this instability. The review presents modern theoretical approaches to the study of this baroclinic instability. These include the description of the instability in terms of the interaction of the edge Rossby waves, the study of the problem of the optimal perturbations, namely the perturbations with a maximum rate of growth of energy or other functionals, the analysis of the non-linear dynamics of disturbances using low-mode approximations of the Galerkin method. The classical energy criteria for the stability of the 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.