In this paper, we introduce and investigate new concepts of L-weakly and M-weakly demicompact operators. Let E be a Banach lattice. An operator T : E −→ E is called L-weakly demicompact, if for every norm bounded sequence (x n) in B E such that {x n − Tx n , n ∈ N} is an L-weakly compact subset of E, we have {x n , n ∈ N} is an L-weakly compact subset of E. Additionally, an operator T : E −→ E is called M-weakly demicompact if for every norm bounded disjoint sequence (x n) in E such that ∥x n − Tx n ∥ → 0, we have ∥x n ∥ → 0. L-weakly (resp. M-weakly) demicompact operators generalize known classes of operators which are L-weakly (resp. M-weakly) compact operators. We also elaborate some properties of these classes of operators.