This textbook is intended for both undergraduate and graduate courses in meteorology and atmospheric sciences, as well as for researchers working on theoretical and numerical aspects of weather and climate or on geophysical fluid dynamics. The treatment is concise, thorough, and self-contained. All necessary concepts are introduced, and the reader is given explicit guidance on all mathematical steps. The book begins with a derivation of the equations of motion. These are then used to discuss fundamental aspects of weather and climate. The mechanisms behind vortical motions, that are known from the daily weather map, are discussed. Shallow-water theory is introduced as a tool for an efficient analysis of key concepts, such as atmospheric waves and synoptic-scale vortices. Quasigeostrophic theory is described and then used to explain the occurrence and mechanisms of extratropical weather by means of baroclinic instability. The specific properties of the atmospheric boundary layer are discussed, with a focus on the interaction between turbulence and mean flows. This is followed by a detailed look at the global atmospheric circulation, highlighting its control by Rossby waves and gravity waves. At the same time, the reader is introduced to essential concepts that find applications in the field, such as balance by geostrophic and hydrostatic equilibrium, the role of entropy and potential temperature, potential vorticity, the Kelvin theorem, instability theory, the Reynolds equations, Eliassen-Palm and pseudo-momentum flux, multi-scale asymptotics, WKB theory, wave action, the transformed Eulerian mean, critical layers, and wave refraction. The text is supplemented by appendices on important mathematical concepts and further elaborations of the main text. Chapter summaries and reading recommendations help the reader not merely to keep focus on the essentials, but just as well to broaden the horizon.