Height restricted resolution (proofs or refutations) is a natural restriction of resolution where the height of the corresponding proof tree is bounded. Height restricted resolution does not distinguish between tree- and sequence-like proofs. We show that polylogarithmic-height resolution is strongly connected to the bounded arithmetic theory S^1_2(\alpha) . We separate polylogarithmic-height resolution from quasi-polynomial size tree-like resolution. Inspired by this we will study infinitely many sub-linear-height restrictions given by functions n → 2_i((log^(i+1) n)^O(1)) for i ≥ 0 . We show that the resulting resolution systems are connected to certain bounded arithmetic theories, and that they form a strict hierarchy of resolution proof systems. To this end we will develop some proof theory for height restricted proofs.