Power Transformer is the heart of the electric power system. One of the main causes of power transformers' insulation failure is Partial discharge (PD). Partial discharges are electrical sparks which occur within the insulation material of power transformers. Degradation of insulation material is indicated by the presence of partial discharges. PD detection in transformers is an intricate task due to the inductive behavior of the transformer windings. The two main categories of PD detection are on-line detection techniques and off-line detection techniques. Off-line PD detection techniques incur high costs due to the downtime of power transformers alongside the utilization of expensive instrumentation. In contrast, on-line PD detection techniques are less expensive and can be performed on energized power transformers. Due to the variable nature of PD activity in power transformers, continuous on-line monitoring of PD activity is an optimal solution for gaining deeper insights into the characteristics of the PD activity. This paper provides an overview of PD activity in power transformers including its mechanism and mathematical modeling. Off-line and on-line PD detection techniques are reviewed and compared.