At first we identify the main error in the formulation of the concept of the weak solution to Navier-Stokes (NS) equations which is the completely insufficient treatment of the incompressibility condition on the fluid (expressed in the standard way by div u = 0). The repair requires the complete reformulation of the NS problem. The basic concept must be the generalized motion (i.e. the generalized flow) which replaces the standard velocity field. Here we define the generalized flow on the bases of Geometric measure theory extended to the theory of Cartesian currents and weak diffeomorphisms (see [1], [2]). Then the key concept of the complete weak solution to the NS problem is defined and the two conjectures (the existence and the regularity ones) concerning the complete weak solutions are formulated. In two appendices many technical details are described (concerning e.g. Cartesian currents, homology conditions, weak diffeomorphisms, etc.). Our approach is based on the unification of the standard analysis of NS equations with the methods of Geometric measure theory and of the theory of Cartesian currents.