A holographic multiple-input multiple-output (MIMO)-based integrated sensing and communications (ISAC) framework is proposed for both downlink and uplink scenarios. The spatial correlation is incorporated into the communication channel modeling, while a spherical wave-based model is used to characterize the sensing link. By considering both instantaneous and statistical channel state information, closed-form expressions are derived for sensing rates (SRs), communication rates (CRs), and outage probabilities under various ISAC designs. This enables an investigation into the theoretical performance limits of the proposed holographic MIMO-based ISAC (HISAC) framework. Further insights are gained by examining the high signal-to-noise ratio (SNR) slopes and diversity orders. Specifically: i) for the downlink case, a sensing-centric (S-C) design and a communications-centric (C-C) design are investigated using different beamforming strategies, and a Pareto optimal design is proposed to characterize the attainable SR-CR region; ii) for the uplink case, the S-C design and the C-C design differ in the interference cancellation order between the communication and sensing signals, with the rate region obtained through a time-sharing strategy. Numerical results are provided to demonstrate that HISAC systems outperform both conventional MIMO-based ISAC systems and holographic MIMO-based frequency-division sensing and communications systems, underscoring the superior performance of the HISAC framework.