  The Fast Diffusion Equation (FDE) $u_t= \Delta u^m$, with $m\in (0,1)$, is an
important model for singular nonlinear (density dependent) diffusive phenomena.
Here, we focus on the Cauchy-Dirichlet problem posed on smooth bounded
Euclidean domains. In addition to its physical relevance, there are many
aspects that make this equation particularly interesting from the pure
mathematical perspective. For instance: mass is lost and solutions may
extinguish in finite time, merely integrable data can produce unbounded
solutions, classical forms of Harnack inequalities (and other regularity
estimates) fail to be true, etc.
  In this paper, we first provide a survey (enriched with an extensive
bibliography) focussing on the more recent results about existence, uniqueness,
boundedness and positivity (i.e., Harnack inequalities, both local and global),
and higher regularity estimates (also up to the boundary and possibly up to the
extinction time). We then prove new global (in space and time) Harnack
estimates in the subcritical regime. In the last section, we devote a special
attention to the asymptotic behaviour, from the first pioneering results to the
latest sharp results, and we present some new asymptotic results in the
subcritical case.
