This is an excellent review of key unsolved problems of mathematical fluid dynamics and of their current state. These problems concern global existence and uniqueness theorems for basic boundary and initial-boundary value problems in the theory of ideal and viscous incompressible fluids.
Contents: 1. Introduction; 2. Mathematical models of hydrodynamics; 3. Uniqueness, global existence and nonexistence of the solution; 4. General stability theory for viscous fluid flows; 5. Stability of ideal fluid flows; 6. Stability of the simplest laminar flows and the first transition; 7. Transitions and chaotic regimes; 8. Asymptotics of vanishing viscosity and turbulence.
The bibliography contains 52 items.