Several new multiuser detectors are developed for different direct-sequence codedivision multiple-access (DS-CDMA) application environments. The first detector is based on a semidefinite-programming (SDP) relaxation technique. In this detector, maximum likelihood (ML) detection is achieved by 'relaxing' the associated combinatorial problem into an SDP problem, which leads to a detector of polynomial complexity. It is shown that the SDP-relaxation (SDPR) based detector can be obtained by solving a dual SDP problem which leads to improved efficiency. Computer simulations demonstrate that the SDPR detector offers near-optimal performance with much reduced computational complexity compared with that of the ML detector proposed by Verdu for both synchronous and asynchronous DS-CDMA systems. The second detector is based on a recursive convex programming (RCP) approach. In this detector, ML detection is carried out in two steps: first, the combinatorial problem associated with ML detection is relaxed to a convex programming problem, and then a recursive approach is used to obtain an approximate solution for ML detection. Efficient unconstrained relaxation approach is proposed for the proposed detector to reduce the involved computational complexity. Computer simulations demonstrate that the proposed detectors offer near-optimal detection performance which is superior to that offered by many other suboptimal detectors including the SDPR detector. However, the computational complexity involved in the proposed detectors is much lower relative to that involved in Verdu's ML detector as well as our SDPR detector. The third detector entails a subspace estimation-based constrained optimization approach for channel estimation in DS-CDMA systems with multipath propagation channels. The proposed approach offers an improved approximation for the noise iii subspace compared with that offered by several existing algorithms. Computer simulations show that the performance of the proposed detector offers nearly the same performance as that of existing subspace detectors but leads to a significant reduction in the amount of computation. Relative to some existing constrained optimization methods, the proposed detector offers a significantly improved performance while requiring a comparable amount of computation. The fourth detector is proposed based on a vector constant-modulus (VCM) approach. This detector is designed for DS-CDMA systems with multipath propagation channels where the effective signatures observed at receiver are distorted by multipath propagation and aliasing concurrently. In this detector, detection is carried out by solving a linear constrained optimization problem whose objective function is formulated based on the VCM criterion. Two adaptation algorithms, namely, the constrained stochastic gradient algorithm and the recursive vector constant-modulus algorithm, are developed. Analysis are presented to investigate the performance of the proposed detector. Computer simulations show that the proposed detectors are able to suppress multiuser interference and inter-symbol interference effectively. More importantly, they offer robust detection performance against the effective signature distortion caused by aliasing at the receiver.