  In this survey paper, I discuss some recent progress on the existence and
regularity of Brakke flows. These include: an "end-time version" of Brakke's
local regularity theorem, which allows to extend the validity of the celebrated
regularity theorem by White from limits of smooth mean curvature flows to
arbitrary Brakke flows; a global-in-time existence theorem for multi-phase
Brakke flows of grain boundaries satisfying suitable ${\rm BV}$ regularity in
time, in both the unconstrained and the fixed boundary settings, with
applications to Plateau's problem for the latter; and the proof that branching
singularities of minimal surfaces are a trigger for dynamical instability, in
the sense that they may be "perturbed away" by a non-trivial canonical Brakke
flow. This note is an extended version of a talk given by the author at the
MATRIX Research Institute on the occasion of the workshop entitled "Minimal
surfaces and geometric flows: interaction between the local and the nonlocal
worlds".
