The study of quasi-randomness is a flourishing topic on uniform hypergraphs. F. R. K. Chung and R. L. Graham (among others) investigated thoroughly quasi-random uniform hypergraphs of density 1/2, showing a series of important equivalent statements about these structures. In this investigations the notion of deviation plays a central role. Chung and Graham explicitly asked to extend these investigations to the case where the fixed density differs from 1/2. This extensive paper starts this investigation. Its main aim is to show a series of relevant equivalent statements about quasi-random uniform hypergraphs of fixed (but not 1/2) density. The authors developed a new technique based on the notion of discrepancy and their delicate proofs are quite different from the ones used by Chung and Graham.