These lectures can be regarded as an Introduction to the Mathematical Theory of the Navier-Stokes equations, addresses to undergraduate and postgraduate students who would like to learn more about one of the most important equations of the mathematical physics. The main point of the discussions will be: the existence of weak or strong solutions to the initial boundary value problems for the Navier-Stokes equations, weak-strong uniqueness theorems, the partial regularity of weak solutions, etc. It would be helpful to have a preliminary knowledge about the theory of elliptic and parabolic equations although some points important for the NavierStokes equations will be explained in lectures.The material based on courses given at Mathematical Institute of Oxford University.

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