In 1982 Beilinson, Bernstein and Deligne have proved that the heart
H(X,Y) of a faithful torsion pair (X,Y) is an abelian category. In 2009
Colpi and Gregorio have proved that H(X,Y) is a Grothendieck category
if and only if the torsion free class Y is cogenerated by a 1-cotilting
module. Recently in a joint paper with Colpi and Mantese we have found
necessary and sufficient conditions for H(X,Y) to be equivalent to a
module category. In this talk I will describe the results obtained.