The paper considers various iterative detection processes that are suitable for use in a synchronous serial data-transmission system in which a q.a.m. signal is fed over a telephone circuit or h.f. radio link at a transmission rate of up to 20 000 bits/s. Reasons are first given for the study of q.a.m. systems. The methods of operation of the different detection processes are then described, and, finally, the results of computer simulation tests are presented, comparing the tolerances of the detection processes to additive white Gaussiannoise with the tolerance of a nonlinear equaliser of optimum design. Several different time-invariant channels are used in the tests. It is shown that a relatively simple detection process can achieve a useful improvement in tolerance to noise over the nonlinear equaliser.