Turbulence, a phenomenon observed by physicists in natural and laboratory flows, is thought to be described by Navier-Stokes equations (NSE). Yet, it is not yet known whether the Cauchy problem is well posed for these equations, and whether solutions of finite energy are regular or unique. In the physics community, the "singularity hypothesis" is generally discarded, following the reasonable principle that "singularities are a mathematical curiosity, in nature they do not exist".

In these lectures, I will discuss how recent progresses in mathematics, numerical simulations and laboratory experiments changed such simple vision leading to a new picture where quasi-singularities living beyond Kolmogorov scale play a central role.

Topics covered:

• The basic empirical laws of turbulence
• Some mathematical aspects of Navier-Stokes equations
• Symmetries and their consequences
• Singularities and anomalous laws
• Predictability of turbulence