In this paper, let （R, m） be a Noetherian local ring, I lohtain in R an ideal, M and N be two finitely generated modules. Firstly, we study the properties of HI^t（M）, t = f-depth（I, M） and discuss the relationship between the Artinianness of HI^i（M, N） and the Artinianness of HI^i（N）. Then, we get that HI^d（M, N） is I-cofinite, if （R, m） is a d-dimensional Gorenstein local ring.
Let F be a Hilbert filtration with respect to a Cohen-Macaulay module M. When G（F, M） and FK（F,M） have almost maximal depths, the Hilbert coefficients gi（F, M） is calculated. In the general case, an upper bound for g2（F, M） is also given.
