A number of initial-boundary-value problems for the equation of fast diffusion are analysed (at varying levels of detail and completeness), i.e., $$\frac{\partial u}{\partial t} = \nabla \cdot (u^{-n} \nabla u)$$ with n > 0, in dimension N > 2 and with zero-Dirichlet boundary data, namely (i) the Cauchy problem (no boundary), mainly summarising existing results, (ii) the interior problem for a simply connected bounded domain (in large part revisiting earlier results), (iii) the problem exterior to a simply connected bounded domain and (iv) the half-space problem (for which we include N =2). The critical (borderline) case \({n = n_{s} \equiv 4/(N+2)}\) , which arises in Yamabe flow, is the subject of particular focus, in part because it provides considerable insight into both the subcritical case, 0 < n < ns, and the supercritical one, ns < n < 1. The results are based on formal-asymptotic analysis and suggest a range of conjectures that could be the subject of rigorous studies. The role of distinct types of similarity solutions is highlighted.