A photodetector converts optical signals to detectable electrical signals. Lately, self-powered photodetectors have been widely studied because of their advantages in device miniaturization and low power consumption, which make them preferable in various applications, especially those related to green technology and flexible electronics. Since self-powered photodetectors do not have an external power supply at zero bias, it is important to ensure that the built-in potential in the device produces a sufficiently thick depletion region that efficiently sweeps the carriers across the junction, resulting in detectable electrical signals even at very low-optical power signals. Therefore, two-dimensional (2D) materials are explored as an alternative to silicon-based active regions in the photodetector. In addition, plasmonic effects coupled with self-powered photodetectors will further enhance light absorption and scattering, which contribute to the improvement of the device's photocurrent generation. Hence, this review focuses on the employment of 2D materials such as graphene and molybdenum disulfide (MoS2) with the insertion of hexagonal boron nitride (h-BN) and plasmonic nanoparticles. All these approaches have shown performance improvement of photodetectors for self-powering applications. A comprehensive analysis encompassing 2D material characterization, theoretical and numerical modelling, device physics, fabrication and characterization of photodetectors with graphene/MoS2 and graphene/h-BN/MoS2 heterostructures with plasmonic effect is presented with potential leads to new research opportunities.