Mathematics of two dimensional turbulence This book is dedicated to the mathematical study of two dimensional statistical hydrodynamics and turbulence described by the 2D Navier Stokes system with a random force. The authors main goal is to justify the statistical properties of a fluids velocity field u(t x) that physicists assume in their work. They rigorously prove that u(t x) converges as time grows to a statistical equilibrium independent of initial data. They use this to study ergodic properties of u(t x) proving in particular that observables f(u(t .)) satisfy the strong law of large numbers and central limit theorem. They also discuss the inviscid limit when viscosity goes to zero normalising the force so that the energy of solutions stays constant while their Reynolds numbers grow to infinity. They show that then the statistical equilibria converge to invariant measures of the 2D Euler equation and study these measures. The methods apply to other nonlinear PDEs perturbed by random forces This book deals with basic problems and questions interesting for physicists and engineers working in the theory of turbulence. Accordingly Chapters 3 5 (which form the main part of this book) end with sections where we explain the physical relevance of the obtained results. These sections also provide brief summaries of the corresponding chapters. In Chapters 3 and 4 our main goal is to justify for the 2D case the statistical properties of fluids velocity