Let X be a ball quasi-Banach function space satisfying some mild additional assumptions and Hx(Rn) the associated Hardy-type space.In this article,we first establish the finite atomic characterization of Hx (Rn).As an application,we prove that the dual space of Hx (Rn) is the Campanato space associated with X.For any given α ∈ (0,1]and s ∈ Z+,using the atomic and the Littlewood-Paley function characterizations of Hx(Rn),we also establish its s-order intrinsic square function characterizations,respectively,in terms of the intrinsic Lusin-area function Sα,s,the intrinsic g-function gα,s,and the intrinsic g*λ-function g*λ,α,s,where λ coincides with the best known range.