A continuous-aperture array (CAPA)-based integrated sensing and communications (ISAC) framework is proposed for both downlink and uplink scenarios. Within this framework, continuous operator-based signal models are employed to describe the sensing and communication processes. The performance of communication and sensing is analyzed using two information-theoretic metrics: the communication rate (CR) and the sensing rate (SR). 1) For downlink ISAC, three continuous beamforming designs are proposed: i) the communications-centric (C-C) design that maximizes the CR, ii) the sensing-centric (S-C) design that maximizes the SR, and iii) the Pareto-optimal design that characterizes the Pareto boundary of the CR-SR region. A low-complexity signal subspace-based approach is proposed to derive the closed-form optimal beamformers for the considered designs. On this basis, closed-form expressions are derived for the achievable CRs and SRs, and the downlink rate region achieved by CAPAs is characterized. 2) For uplink ISAC, the C-C and S-C successive interference cancellation-based methods are proposed to manage inter-functionality interference. Using the subspace approach closed-form expressions for the optimal detectors as well as the achievable CRs and SRs are derived. The uplink SR-CR region is characterized based on the time-sharing technique. Numerical results demonstrate that, for both downlink and uplink, CAPA-based ISAC achieves higher CRs and SRs as well as larger CR-SR regions compared to conventional spatially discrete array-based ISAC.