The capacity limits of continuous-aperture array (CAPA)-based wireless communications are characterized. To this end, an analytically tractable transmission framework is established for both uplink and downlink CAPA systems. Based on this framework, closed-form expressions for the single-user channel capacity are derived. The results are further extended to a multiuser case by characterizing the capacity limits of a two-user channel and proposing the associated capacity-achieving decoding and encoding schemes. In the uplink case, the capacity-achieving detectors and sum-rate capacity are derived, and the capacity region is characterized. In the downlink case, the uplink-downlink duality is established by deriving the uplink-to-downlink and downlink-to-uplink transformations under the same power constraint, based on which the optimal source current distributions and the achieved sum-rate capacity and capacity region are characterized. For comparison, the uplink and downlink sum-rates achieved by the linear zero-forcing scheme are also analyzed. To gain further insights, several case studies are presented by specializing the derived results into various array structures, including the planar CAPA, linear CAPA, and planar spatially discrete array (SPDA). Numerical results are provided to reveal that the channel capacity achieved by CAPAs converges towards a finite upper bound as the aperture size increases; and CAPAs offer superior capacity over the conventional SPDAs.