This seminar will focus on the dynamical-systems approach to turbulence: rather than attempting to predict an individual turbulent trajectory for long times, one asks whether turbulent statistics and recurrent patterns can be organized by unstable invariant solutions—equilibria, traveling waves, periodic orbits, and relative periodic orbits—embedded in the turbulent state space. This viewpoint has been developed in fluid dynamics by CvitanoviÅLc and collaborators, Kawahara–Kida, Kawahara–Uhlmann–van Veen, Chandler–Kerswell, Willis–CvitanoviÅLc–Avila, Page–Kerswell, and others. A central methodological point of the seminar is that we will not try to prove that the Navier–Stokes flow itself is Anosov, nor assume that fully developed turbulence is uniformly hyperbolic. That would be far stronger than what is presently justified. Instead, uniformly hyperbolic dynamics will serve as a mathematically controlled model problem. For an Anosov flow, periodic orbits, transfer operators, dynamical zeta functions, and Pollicott–Ruelle resonances fit into a rigorous spectral theory. The question for turbulence is then: Can an effective hyperbolic/Anosov description capture the dynamically relevant part of a turbulent attractor?