  In this article we initiate a systematic study of irreducible weight modules
over direct limits of reductive Lie algebras, and in particular over the simple
Lie algebras $A(\infty)$, $B(\infty)$, $C(\infty)$ and $D(\infty)$. Our main
tool is the shadow method introduced recently in \cite{DMP}. The integrable
irreducible modules are an important particular class and we give an explicit
parametrization of the finite integrable modules which are analogues of
finite-dimensional irreducible modules over reductive Lie algebras. We then
introduce the more general class of pseudo highest weight modules. Our most
general result is the description of the support of any irreducible weight
module.
