Aperiodic quadriphase Z-complementary sequences, which include the conventional complementary sequences as special cases, are introduced. It is shown that, the aperiodic quadriphase Z-complementary pairs are normally better than binary ones of the same length, in terms of the number of Z-complementary pairs, and the maximum zero correlation zone. New notions of elementary transformations on quadriphase sequences and elementary operations on sets of quadriphase Z-complementary sequences are presented. In particular, new methods for analyzing the relations among the formulas relative to sets of quadriphase Z-complementary sequences and for describing the sets are proposed. The existence problem of Z-complementary pairs of quadriphase sequences with zero correlation zone equal to 2, 3, and 4 is investigated. Constructions of sets of quadriphase Z-complementary sequences and their mates are given.