The Introductory Chapter -- Dynamics of Discrete Vortices -- Dynamics of Finite-core Vortices -- The Concluding Chapter. This book presents an extensive analysis of the dynamics of discrete and distributed baroclinic vortices in a multi-layer fluid that characterizes the main features of the large and mesoscales dynamics of the atmosphere and the ocean. It widely covers the case of hetonic situations as well as the case of intrathermocline vortices that are familiar in oceanographic and of recognized importance for heat and mass transfers. Extensive typology of such baroclinic eddies is made and analysed with the help of theoretical development and numerical computations. As a whole it gives an overview and synthesis of all the many situations that can be encountered based on the long history of the theory of vortex motion and on many new situations. It gives a renewed insight on the extraordinary richness of vortex dynamics and open the way for new theoretical, observational and experimental advances. This volume is of interest to experts in physical oceanography, meteorology, hydrodynamics, dynamic systems, involved in theoretical, experimental and applied research and lecturers, post-graduate students, and students in these fields.