  We review two important non-perturbative approaches for extracting the
physics of low-dimensional strongly correlated quantum systems. Firstly, we
start by providing a comprehensive review of non-Abelian bosonization. This
includes an introduction to the basic elements of conformal field theory as
applied to systems with a current algebra, and we orient the reader by
presenting a number of applications of non-Abelian bosonization to models with
large symmetries. We then tie this technique into recent advances in the
ability of cold atomic systems to realize complex symmetries. Secondly, we
discuss truncated spectrum methods for the numerical study of systems in one
and two dimensions. For one-dimensional systems we provide the reader with
considerable insight into the methodology by reviewing canonical applications
of the technique to the Ising model (and its variants) and the sine-Gordon
model. Following this we review recent work on the development of
renormalization groups, both numerical and analytical, that alleviate the
effects of truncating the spectrum. Using these technologies, we consider a
number of applications to one-dimensional systems: properties of carbon
nanotubes, quenches in the Lieb-Liniger model, 1+1D quantum chromodynamics, as
well as Landau-Ginzburg theories. In the final part we move our attention to
consider truncated spectrum methods applied to two-dimensional systems. This
involves combining truncated spectrum methods with matrix product state
algorithms. We describe applications of this method to two-dimensional systems
of free fermions and the quantum Ising model, including their non-equilibrium
dynamics.
