We introduce the concept of strong BD property in Banach lattices and we characterize Banach lattices with this property. Also, by introducing the class of almost limited (resp. Dunford-Pettis) weakly completely continuous operators from an arbitrary Banach lattice E to another F, we give some properties of them related to some well known classes of operators and related to the strong BD (resp. RDP *) property of the Banach lattice E.