The carrier-frequency offset effect on the performance of asynchronous multicarrier direct-sequence code-division multiple-access (MC-DS-CDMA) systems with aperiodic random spreading and correlated Rayleigh fading is studied in this paper. We obtain the optimum combining filter that maximizes the signal-to-interference-plus-noise ratio (SINR) of the combined statistics and exploits correlated information among subchannels. A closed-form expression for the unconditional covariance matrix of the interference-plus-noise vector, which forms the basis of our theoretical analysis of the maximum SINR and the average bit error probability formula, is derived by averaging several random parameters including asynchronous delays, correlated Rayleigh fading, and signature sequences. The analytic results obtained are applicable to MC-CDMA with appropriate modifications. Furthermore, we show that the MC-CDMA system with a common random signature sequence over all subcarriers for a given user outperforms that with distinct sequences over different subcarriers. Finally, the performance of MC-CDMA systems using the optimum combining technique is compared with that of different combining filters in the simulation.