  Turbulence is the most important, ubiquitous, and difficult problem of
classical physics. Feynman viewed it as essentially unsolved, without a
rigorous mathematical basis to describe the statistical dynamics of this most
complex of fluid motion. However, the paradigm shift came in 1959, with the
formulation of the Eulerian direct interaction approximation (DIA) closure by
Kraichnan. It was again based on renormalized perturbation theory, like quantum
electrodynamics, and is a bare vertex theory that is manifestly realizable.
Here, we review some of the subsequent exciting achievements in closure theory.
We also document in some detail the progress that has been made in extending
statistical dynamical turbulence theory to the real world of interactions with
mean flows, waves and inhomogeneities such as topography. This includes
numerically efficient inhomogeneous closures, like the realizable
quasi-diagonal direct interaction approximation (QDIA), and even more efficient
Markovian inhomogeneous closures (MICs). Recent developments include the
formulation and testing of an eddy damped Markovian anisotropic closure (EDMAC)
that is realizable in interactions with transient waves but is as efficient as
the eddy damped quasi-normal Markovian (EDQNM). As well a similarly efficient
closure, the realizable eddy damped Markovian inhomogeneous closure (EDMIC) has
been developed. Moreover, we present subgrid models that cater for the complex
interactions that occur in geophysical flows. Recent progress includes the
determination of complete sets of subgrid terms for skilful large eddy
simulations of baroclinic inhomogeneous turbulent atmospheric and oceanic flows
interacting with Rossby waves and topography. The success of these
inhomogeneous closures has also led to further applications in data
assimilation and ensemble prediction and generalization to quantum fields.
