The HRS tilting process and Grothendieck hearts of t-structures

In this paper we revisit the problem of determining when the heart of a t-structure is a Grothendieck category, with special attention to the case of the Happel-Reiten-Smal{\o} (HSR) t-structure in the derived category of a Grothendieck category associated to a torsion pair in the latter. We revisit the HRS tilting process deriving from it a lot of information on the HRS t-structures which have a projective generator or an injective cogenerator, and obtain several bijections between classes of pairs $(\mathcal{A},\mathbf{t})$ consisting of an abelian category and a torsion pair in it. We use these bijections to re-prove, by different methods, a recent result of Tilting Theory and the fact that if $\mathbf{t}=(\mathcal{T},\mathcal{F})$ is a torsion pair in a Grothendieck category $\mathcal{G}$, then the heart of the associated HRS t-structure is itself a Grothendieck category if, and only if, $\mathbf{t}$ is of finite type. We survey this last problem and recent results after its solution.