Code-division multiple access (CDMA) is the basis of a family of advanced air interfaces in current and future generation networks. The benefits promised by CDMA have not been fully realized partly due to the prohibitive complexity of optimal detection and decoding of many users communicating simultaneously using the same frequency band. From both theoretical and practical perspectives, this paper advocates a new paradigm of CDMA with sparse spreading sequences, which enables near-optimal multiuser detection using belief propagation (BP) with low-complexity. The scheme is in part inspired by capacity-approaching low-density parity-check (LDPC) codes and the success of iterative decoding techniques. Specifically, it is shown that BP-based detection is optimal in the large-system limit under many practical circumstances, which is a unique advantage of sparsely spread CDMA systems. Moreover, it is shown that, from the viewpoint of an individual user, the CDMA channel is asymptotically equ...