This paper addresses the capacity limits of continuous-aperture array (CAPA)-based wireless communications, presenting an analytically tractable transmission framework for both uplink and downlink CAPA systems. The authors derive closed-form expressions for the single-user channel capacity and extend the analysis to multiuser scenarios, focusing on a two-user channel. For the uplink, the sum-rate capacity and capacity region are derived, along with optimal detectors. For the downlink, the uplink-downlink duality is established, and the optimal power allocation policy and sum-rate capacity are characterized. Case studies for planar CAPA, linear CAPA, and planar spatially discrete array (SPDA) are presented, revealing that CAPA offers significant capacity gains over conventional SPDA as the aperture size increases. Numerical results demonstrate that CAPA's capacity converges to a finite upper bound and outperforms SPDA in terms of capacity.This paper addresses the capacity limits of continuous-aperture array (CAPA)-based wireless communications, presenting an analytically tractable transmission framework for both uplink and downlink CAPA systems. The authors derive closed-form expressions for the single-user channel capacity and extend the analysis to multiuser scenarios, focusing on a two-user channel. For the uplink, the sum-rate capacity and capacity region are derived, along with optimal detectors. For the downlink, the uplink-downlink duality is established, and the optimal power allocation policy and sum-rate capacity are characterized. Case studies for planar CAPA, linear CAPA, and planar spatially discrete array (SPDA) are presented, revealing that CAPA offers significant capacity gains over conventional SPDA as the aperture size increases. Numerical results demonstrate that CAPA's capacity converges to a finite upper bound and outperforms SPDA in terms of capacity.