Adaptive iterative receivers which combine multiuser decision-feedback detection with maximum a posteriori (MAP) decoding and soft feedback are presented for synchronous coded direct sequence-code-division multiple access. Both successive and parallel demodulation of users are considered. Optimal filters are derived using both minimum mean squared error and least squares (LS) criteria. The latter assumes short (repeated) spreading codes and that the users to be demodulated simultaneously transmit training sequences. The LS criterion does not require prior knowledge or estimates of spreading codes and channels. Simulation results show that the adaptive receiver can perform significantly better than the standard (soft) interference canceller, since the adaptive algorithm attempts to measure and exploit the second-order statistics between the input and output of the MAP decoder. With limited training, successive feedback and decoding performs significantly better than parallel feedback. The effect of code rate on performance is examined, and reduced-rank versions of the adaptive LS algorithms, which can reduce training overhead, are also presented.