  The core of an ideal is defined as the intersection of all of its reductions.
In this paper we provide an explicit description for the core of a monomial
ideal $I$ satisfying certain residual conditions, showing that ${\rm core}(I)$
coincides with the largest monomial ideal contained in a general reduction of
$I$. We prove that the class of lex-segment ideals satisfies these residual
conditions and study the core of lex-segment ideals generated in one degree.
For monomial ideals that do not necessarily satisfy the residual conditions and
that are generated in one degree, we conjecture an explicit formula for the
core, and make progress towards this conjecture.
