  We present a setting for the study of torsion theories in general categories.
The idea is to associate, with any pair ($\mathcal T$, $\mathcal F$) of full
replete subcategories in a category $\mathcal C$, the corresponding full
subcategory $\mathcal Z = \mathcal T \cap \mathcal F$ of \emph{trivial objects}
in $\mathcal C$. The morphisms which factor through $\mathcal Z$ are called
$\mathcal Z$-trivial, and these form an ideal of morphisms, with respect to
which one can define $\mathcal Z$-prekernels, $\mathcal Z$-precokernels, and
short $\mathcal Z$-preexact sequences. This naturally leads to the notion of
pretorsion theory, which is the object of study of this article, and includes
the classical one in the abelian context when $\mathcal Z$ is reduced to the
$0$-object of $\mathcal C$. We study the basic properties of pretorsion
theories, and examine some new examples in the category of all endomappings of
finite sets and in the category of preordered sets.
