The Boltzmann equation (BE) is the keystone of the kinetic theory of fluids, describing flow in the transitional molecular/continuum regime. BE provides an evolution equation for the so-called one-particle marginal, viz., the probability density of particles in the position/velocity space. Accordingly, BE is high-dimensional (2D+1 for D spatial dimensions, where +1 accounts for time dependence). The BE has several fundamental structural properties, notably, certain invariance principles and decay of an entropy functional (the celebrated H theorem). These structural properties underly the connection between BE and conventional continuum models, and it can be shown that (BE) encapsulates all conventional continuum models, such as the Navier-Stokes-Fourier system, as limit solutions. Accordingly, (BE) inherently corresponds to a multiscale model.

The vast majority of numerical techniques for kinetic equations is based on stochastic particle methods. From a rigorous approximation perspective, such particle methods have several shortcomings. A relatively sparsely investigated approach, is provided by moment-closure systems. Such moment-closure approximation inherently exploit the structural properties of (BE) to arrive at efficient approximations. One of the fundamental properties of moment-closure systems, is that they yield a natural  hierarchy, which makes them ideally suited for (goal-)adaptive approximations. In this presentation, I will outline the basic elements of a goal-adaptive moment-closure method for (BE), and the recent progress that we have towards the development of such a methodology. The presentation ends with some open problems.