  We introduce the class of unbounded $M$-weakly operators and the class of
unbounded $L$-weakly compact operators. We investigate some properties for
these new classification of operators and we study relation between them and
$M$-weakly compact and $L$-weakly compact operators. We also present an
operator characterization of Banach lattices with order continuous norm.
\keywords{unbounded $M$-weakly compact \and unbounded $L$-weakly compact \and
unbounded norm convergence \and $M$-weakly compact \and $L$-weakly compact
