  We study the Cauchy problem for the advection-diffusion equation $\partial_t
u + \mathrm{div} (u b ) = \Delta u$ associated with a merely integrable
divergence-free vector field $b$ defined on the torus. We discuss existence,
regularity and uniqueness results for distributional and parabolic solutions,
in different regimes of integrability both for the vector field and for the
initial datum. We offer an up-to-date picture of the available results
scattered in the literature, and we include some original proofs. We also
propose some open problems, motivated by very recent results which show
ill-posedness of the equation in certain regimes of integrability via convex
integration schemes.
