  This thesis deals with the general problem of determining when the heart
$\mathcal{H}$ of a t-structure in a triangulated category $\mathcal{D}$ is a
Grothendieck or a module category. As preliminaries, we study Grothendieck
conditions AB3-AB5 for $\mathcal{H}$ in a very general setting. We then
concentrate on two familiar examples of smashing t-structures. First, we
consider that $\mathcal{D}=\mathcal{D}(\mathcal{G})$ is the (unbounded) derived
category of a Grothendieck category $\mathcal{G}$ and that the t-structure is
the one associated to a torsion pair $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in
$\mathcal{G}$, usually known as Happel-Reiten-Smal$\emptyset$ t-structure. In
the second example studied, we assume that $\mathcal{D}=\mathcal{D}(R)$ is the
derived category of a commutative Noetherian ring $R$ and that the t-structure
is compactly generated. On what concern the Happel-Reiten-Smal$\emptyset$
example, we show that if $\mathcal{H}=\mathcal{H}_\mathbf{t}$ is AB5, then
$\mathcal{F}$ is closed under taking direct limits in $\mathcal{G}$. Moreover,
the converse is true, even implying that $\mathcal{H}_\mathbf{t}$ is a
Grothendieck category, for a wide class of torsion pairs in $\mathcal{G}$ which
includes the hereditary, tilting and cotilting ones. When
$\mathcal{G}=R-\text{Mod}$ is a module category, we are able to identify the
hereditary torsion pairs $\mathbf{t}$ in $R-\text{Mod}$ for which
$\mathcal{H}_\mathbf{t}$ is a module category. When $R$ is a commutative
noetherian ring, we show that all compactly generated t-structures in
$\mathcal{D}(R)$ whose associated filtration by supports is left bounded have a
heart $\mathcal{H}$ which is a Grothendieck category. This is used to identify
all compactly generated t-structures in $\mathcal{D}(R)$ whose heart is a
module category.
