The treatment of prostate cancer (PCa) is shifting towards the use of highly accurate image-guided procedures in order to achieve better oncologic results. A common strategy consists of using both pre-operative multiparametric magnetic resonance imaging and intra-operative transrectal ultrasound during a prostate biopsy or focal ablation procedure, thus offering enhanced localisation information through high spatial resolution and dynamic response, respectively. However, reliable non-rigid registration is still technically challenging owing to differences in cross-modal imaging physics as well as large deformations between the two modalities caused by rectal probe compression; this paper reviews how deep learning has evolved, focusing on convolutional neural networks, generative models, including generative adversarial networks and diffusion models, and transformer-based architectures. We discuss the extent to which they utilise biomechanical priors to inform the solution of registration problems, against standardized challenges such as µ-RegPro. State-of-the-art approaches achieve sub-millimetre target registration errors with real-time inference times for intra-operative deployment. Addressing outstanding challenges related to interpretation and generalization, this review provides an outlook of the road map to develop "physics-aware" smart interventional systems. All these developments represent important steps toward a fully automated, precise, and minimally invasive PCa management pipeline.