  In this paper, we will study some properties of b-weakly compact operators
and we will investigate their relationships to some variety of operators on the
normed vector lattices. With some new conditions, we show that the modulus of
an operator $T$ from Banach lattice $E$ into Dedekind complete Banach lattice
$F$ exists and is $b$-weakly operator whenever $T$ is a $b$-weakly compact
operator. We show that every Dunford-Pettis operator from a Banach lattice $E$
into a Banach space $X$ is b-weakly compact, and the converse holds whenever
$E$ is an $AM$-space or the norm of $E^\prime$ is order continuous and $E$ has
the Dunford-Pettis property. We also show that each order bounded operator from
a Banach lattice into a $KB$-space admits a $b$-weakly compact modulus.
