The monograph focuses on the analytical challenges of describing incompressible fluid flows using Lagrangian variables. The Lagrangian method is presented as a complementary approach to the Eulerian method for studying fluid motion. The fundamental nature of fluid dynamics is linked to tracing the trajectories of individual fluid particles, which is precisely what the Lagrangian method facilitates. The purpose of this publication is to reflect as fully as possible the achievements of analytical fluid dynamics in Lagrangian variables. A significant part of it consists of original articles by the author. Considerable attention is paid to the subtleties of the Lagrangian description and its popularization among experts in fluid dynamics, students, postgraduates, and anyone interested in problems of fluid mechanics. The book can be considered, among other things, as a tutorial on Lagrangian fluid dynamics. The book covers issues of the general description of the movement of incompressible fluid, problems of classical hydrodynamics and geophysical hydrodynamics. Fundamental topics include free-boundary motion, vortices in fluids, and water waves. The book includes a number of problems that take into account the stratification and viscosity of fluids, as well as the rotation of the Earth. Sample Chapter(s) Preface Chapter 7: Two-Dimensional Vortex Flow Contents: Description of Basics: Ideal Fluid Equations Potential Motion Vortex Motion Waves on Water: Potential Approximation Gerstner Wave Weakly Vortex Waves Exact Solutions in Vortex Dynamics: Two-Dimensional Vortex Flow Localized Two-Dimensional Vortices in the External Flow Spatial Vortex Flows Generalized Gerstner Waves: Waves on Water at Unstable Pressure Vortex Model of Rogue Waves Breaking of the Surface Gravity Wave Non-Stationary Edge Waves Viscous Fluid Flows: Basic Equations and Examples Gravity Waves on the Surface of a Viscous Fluid The Earth's Rotation Effect: Exact Solutions for Waves Vortices in a Rotating Fluid Readership: This book can be targeted by a various levels of researchers from undergraduate and beginner to graduate and professional levels for both industry and academic sectors in green and sustainable energy.