The article reviews the principles underlying the use of isentropic maps of potential vorticity (PV) to represent dynamical processes in the atmosphere, including the Lagrangian conservation principle and the invertibility principle. The authors discuss how these principles can be extended to account for lower boundary conditions and diabatic and frictional processes. They illustrate the insights gained from this approach through examples from operational weather analyses and idealized theoretical models, focusing on the structure, origin, and persistence of cutoff cyclones and blocking anticyclones, the physical mechanisms of Rossby wave propagation and instability, and the nonlinear behavior of such waves and instabilities. The article also explores the role of PV in understanding the dynamics of the atmosphere, emphasizing the simplicity and completeness of the resulting picture, which is supported by time sequences of isentropic PV and surface potential temperature charts. The authors highlight the importance of coarse-grain approximations to PV distributions and the limitations of this approach in certain situations, such as equatorial Kelvin modes and rapid advection processes.The article reviews the principles underlying the use of isentropic maps of potential vorticity (PV) to represent dynamical processes in the atmosphere, including the Lagrangian conservation principle and the invertibility principle. The authors discuss how these principles can be extended to account for lower boundary conditions and diabatic and frictional processes. They illustrate the insights gained from this approach through examples from operational weather analyses and idealized theoretical models, focusing on the structure, origin, and persistence of cutoff cyclones and blocking anticyclones, the physical mechanisms of Rossby wave propagation and instability, and the nonlinear behavior of such waves and instabilities. The article also explores the role of PV in understanding the dynamics of the atmosphere, emphasizing the simplicity and completeness of the resulting picture, which is supported by time sequences of isentropic PV and surface potential temperature charts. The authors highlight the importance of coarse-grain approximations to PV distributions and the limitations of this approach in certain situations, such as equatorial Kelvin modes and rapid advection processes.