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Let R be a commutative Noetherian ring, let Φ be a system of ideals of R, let M be a finitely generated R-module, and let t be a nonnegative integer. We first show that a general local cohomology module H Φ p i M p is a finitely generated R-module for all i < t if and only if Ass R H Φ p i M p is a finite set and H Φ p i M p is a finitely generated R p -module for all i < t and all 픭 ∈ Spec(R). Then, as a consequence, we prove that if (R, 픪) is a complete local ring, Φ is countable, and n ∈ ℕ0 is such that Ass R H Φ p h Φ n M M ≥ n is a finite set, then f Φ n M = h Φ n M. In addition, we show that the properties of vanishing and finiteness of general local cohomology modules are equivalent at high points over an arbitrary Noetherian (not necessarily local) ring. For each covariant R-linear functor T from Mod(R) into itself, which has the property of global vanishing on Mod(R) , for any Serre subcategory 풮, and t ∈ ℕ, we prove that ℛiT(R) ∈ 풮 for all i ≥ t if and only if ℛiT(M) ∈ 풮 for any finitely generated R-module M and all i ≥ t. We also obtain some results on general local cohomology modules. [ABSTRACT FROM AUTHOR]