  The main purpose of this expository note is to give a short account of the
recent developments in mathematical wave kinetic theory. After reviewing the
physical theory, we explain the importance of the notion of a scaling law,
which dictates the relation between the asymptotic parameters as the kinetic
limit is taken. This sets some natural limitations on the kinetic approximation
that were not precisely understood in the literature as far as we know. We then
describe our recent and upcoming works that give the first full, mathematically
rigorous, derivation of the wave kinetic theory at the natural kinetic
timescale. The key new ingredient is a delicate analysis of the diagrammatic
expansion that allows to a) uncover highly elaborate cancellations at arbitrary
large order of diagrams, and b) overcome difficulties coming from factorial
divergences in the expansion and the criticality of the problem. The results
mentioned in this note appear in our recent works [16, 17] as well as an
upcoming one in [18].
