Monitoring the partial discharge (PD) activity of power equipment insulation is crucial to ensure uninterrupted power system operation. PD occurrence is highly correlated to weakened insulation strength. If PD occurrences are left unchecked, unexpected insulation breakdowns may occur. The comprehensive PD diagnostic process includes the detection, localization, and classification of PD. Accurate PD source localization is necessary to locate the weakened insulation segment. As a result, rapid and precise PD localization has become the primary focus of PD diagnosis for power equipment insulation. This paper presents a review of different approaches to PD localization, including conventional, machine learning (ML), and deep learning (DL) as a subset of ML approaches. The review focuses on the ML and DL approaches developed in the past five years, which have shown promising results over conventional approaches. Additionally, PD detection using conventional, unconventional, and a PCB antenna designed based on UHF techniques is presented and discussed. Important benchmarks, such as the sensors used, algorithms employed, algorithms compared, and performances, are summarized in detail. Finally, the suitability of different localization techniques for different power equipment applications is discussed based on their strengths and limitations.