This book deals with fluid dynamics of incompressible non-viscous fluids. The main goal is to present an argument of large interest for physics, and applications in a rigorous logical and mathematical setup, therefore avoiding cumbersome technicalities. Classical as well as modern mathematical developments are illustrated in this book, which should fill a gap in the present literature. The book does not require a deep mathematical knowledge. The required background is a good understanding of classical arguments of mathematical analysis, including the basic elements of ordinary and partial differential equations, measure theory and analytic functions, and a few notions of potential theory and functional analysis. The contents of the book begins with the Euler equation, construction of solutions, stability of stationary solutions of the Euler equation. It continues with the vortex model, approximation methods, evolution of discontinuities, and concludes with turbulence.