This book, originating from a seminar held at Oberwolfach in 2022, introduces to state-of-the-art methods and results in the study of free boundary problems which are arising from compressible as well as from incompressible Euler’s equations in general. A particular set of such problems is given by gaseous stars considered in a vacuum (modeled via the compressible Euler equations) as well as water waves in their full generality (seen as recasts of incompressible irrotational Euler equations). This is a broad research area which is highly relevant to many real life problems, and in which substantial progress has been made in the last decade.