We present an introductory overview of several challenging problems in the statistical characterization of turbulence. We provide examples from fluid turbulence in three and two dimensions, from the turbulent advection of passive scalars, turbulence in the one-dimensional Burgers equation, and fluid turbulence in the presence of polymer additives.

In this study, we examine the intriguing connection between turbulence and equilibrium statistical mechanics. There are several recent works which emphasize this connection. Thus in the last few years, the first manifestations of the thermalization, predicted by T D Lee in 1952, was seen and a theoretical understanding of this was developed through detailed studies of finite-dimensional, Galerkin-truncated equations of hydrodynamics. Furthermore, the idea of the Galerkin truncation can be generalized for studying turbulence in non-integer (fractal) dimensions to yield a new, critical dimension with an equilibrium Gibbs state coinciding with a Kolmogorov spectrum. In this paper, we discuss these very exciting and recent developments in turbulence as well as open problems for the future.