Understanding the dynamics of planetary and stellar fluid layers, including atmospheres, oceans, iron cores, convective and radiative zones in stars... remains a tremendous interdisciplinary challenge. Beyond the challenge in fundamental fluid mechanics to understand these flows involving rotation, buoyancy, waves, instabilities, turbulence, at typical scales well beyond our day-to-day experience, a global knowledge of the involved processes is fundamental to a better understanding of the dynamics of celestial bodies. Among the numerous open questions, one can for instance mention: • How and where does the energy of the general ocean circulation cascade from the large climatic scales, where most of it is generated, to the smaller scales, where all of it is dissipated? • What are the relevant driving forces and flow regimes in planetary cores for explaining the generation of a large variety of magnetic fields by dynamo processes? • What are the prevalent force balances and physical mechanisms behind the large-scale features such as Jupiter's Great Red Spot and Jupiter's bands? • How are the various types of waves propagating in stellar interiors generated, and how can they be understood via asteroseismology? Interdisciplinary research in geo- and astrophysical fluid dynamics is also intrinsically multi-method. Indeed, the main obstacle to quantitative modeling and understanding of planetary flows stands in the extreme character of the involved dimensionless parameters. Relevant studies thus rely on the principle of dynamical similitude and scaling laws, sustained by theory, experiments and numerical simulations. Much effort has been devoted to understanding planetary and stellar flows within the various communities of Mechanics, Applied Mathematics, Engineering, Physics, Planetary and Earth Sciences, Astrophysics... While open questions actually rely on the same fundamental concepts and phenomena, lots of progress has been made within each enclosed domain, with only marginal cross- fertilizations. The objective of this CISM School is to go beyond this state, by providing participants with a global introduction and an up-to-date overview of all relevant studies, fully addressing the wide range of involved disciplines and methods. The course will be organized in two parts, each consisting of three chapters. The first part will focus on fundamental aspects of fluid mechanics, including introductory material and current research. Its three chapters will be devoted to waves, instabilities, and turbulence. The second part will focus on applications to topical geo- and astrophysical problems. Its three chapters will be devoted to planetary cores, atmospheres and oceans, and stars. All chapters will be pursued in parallel during the whole week, in order to highlight the close link between the models and their various applications. The target audience for this School is PhD students, postdoctoral and young researchers, in departments of Mechanics, Applied Mathematics, Engineering, Physics, Planetary and Earth Sciences, and Astrophysics. A background in fluid dynamics will be assumed, but no specific knowledge in any of the application domains will be requested. This school takes place within the research project FLUDYCO, supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 681835-FLUDYCO-ERC-2015-CoG).