This paper characterizes the capacity limits of continuous-aperture array (CAPA)-based wireless communications. A transmission framework is established for both uplink and downlink CAPA systems, enabling the derivation of closed-form expressions for single-user channel capacity. The results are extended to multiuser scenarios by analyzing a two-user channel and proposing capacity-achieving decoding and encoding schemes. For the uplink case, sum-rate capacity, capacity region, and capacity-achieving detectors are derived. For the downlink case, uplink-downlink duality is established through transformations under the same power constraint, leading to optimal power allocation and sum-rate capacity characterization. Case studies are presented for various array structures, including planar CAPA, linear CAPA, and planar spatially discrete array (SPDA). Numerical results show that CAPA capacity converges to a finite upper bound as aperture size increases, and CAPAs offer significant capacity gains over conventional SPDAs. The paper proposes a novel analytical framework for CAPA-based communications, using electromagnetic field theories to model signals and spatial responses. This framework enables the characterization of channel capacity for both uplink and downlink communications. The main contributions include the design of optimal detectors and source current distributions for single-user CAPA communications, the extension of analysis to multiuser systems, and the establishment of uplink-downlink duality. The results demonstrate that CAPA capacity increases with aperture size and converges to a finite upper limit, outperforming conventional SPDA systems. The paper provides a comprehensive analysis of CAPA-based communications, offering insights into their performance and potential for future wireless communication systems.This paper characterizes the capacity limits of continuous-aperture array (CAPA)-based wireless communications. A transmission framework is established for both uplink and downlink CAPA systems, enabling the derivation of closed-form expressions for single-user channel capacity. The results are extended to multiuser scenarios by analyzing a two-user channel and proposing capacity-achieving decoding and encoding schemes. For the uplink case, sum-rate capacity, capacity region, and capacity-achieving detectors are derived. For the downlink case, uplink-downlink duality is established through transformations under the same power constraint, leading to optimal power allocation and sum-rate capacity characterization. Case studies are presented for various array structures, including planar CAPA, linear CAPA, and planar spatially discrete array (SPDA). Numerical results show that CAPA capacity converges to a finite upper bound as aperture size increases, and CAPAs offer significant capacity gains over conventional SPDAs. The paper proposes a novel analytical framework for CAPA-based communications, using electromagnetic field theories to model signals and spatial responses. This framework enables the characterization of channel capacity for both uplink and downlink communications. The main contributions include the design of optimal detectors and source current distributions for single-user CAPA communications, the extension of analysis to multiuser systems, and the establishment of uplink-downlink duality. The results demonstrate that CAPA capacity increases with aperture size and converges to a finite upper limit, outperforming conventional SPDA systems. The paper provides a comprehensive analysis of CAPA-based communications, offering insights into their performance and potential for future wireless communication systems.