Let R be a commutative ring with identity and M a unitary R-module, and n > 1 an integer number. As a generalization of the concept of prime submodules, a proper submodule N of M will be called n-almost prime, if for r ∈ R and x ∈ M with rx ∈ N \ (N : M) n−1N, either x ∈ N or r ∈ (N : M). We study n-almost prime submodules, in this paper.