Let r > 1 be an integer. An h-hypergraph H is said to be r-quasi-linear (linear for r = 1) if any two edges of H intersect in 0 or r vertices. In this paper it is shown that r-quasi-linear paths P h,r m of length m > 1 and cycles C h,r m of length m > 3 are chromatically unique in the set of h-uniform r-quasi-linear hypergraphs provided r > 2 and h > 3r − 1.