  This contribution covers the topics presented by the authors at the {\it
``Fundamental Problems of Turbulence, 50 Years after the Marseille Conference
1961"} meeting that took place in Marseille in 2011. It focuses on some of the
mathematical approaches to fluid dynamics and turbulence. This contribution
does not pretend to cover or answer, as the reader may discover, the
fundamental questions in turbulence, however, it aims toward presenting some of
the most recent advances in attacking these questions using rigorous
mathematical tools. Moreover, we consider that the proofs of the mathematical
statements (concerning, for instance, finite time regularity, weak solutions
and vanishing viscosity) may contain information as relevant, to the
understanding of the underlying problem, as the statements themselves.
