  Moir\'e materials provide a highly tunable environment for the realization of
band structures with engineered physical properties. Specifically, moir\'e
structures with Fermi surface flat bands - a synthetic environment for the
realization of correlated phases - have moir\'e unit cells containing thousands
of atoms and tantalizingly complex bands structures. In this paper we show that
statistical principles go a long way in explaining universal physical
properties of these systems. Our approach builds on three conceptual elements:
the presence of quantum chaos caused by the effective irregularity of the
atomic configurations on short length scales, Anderson localization in momentum
space, and the presence of approximate crystalline symmetries. Which of these
principles dominates depends on material parameters such as the extension of
the Fermi surface or the strength of the moir\'e lattice potential. The
phenomenological consequences of this competition are predictions for the
characteristic group velocity of moir\'e bands, a primary indicator for their
average flatness. In addition to these generic features, we identify structures
outside the statistical context, notably almost flat bands close to the extrema
of the unperturbed spectra, and the celebrated zero energy `magic angle' flat
bands, where the latter require exceptionally fine tuned material parameters.
