Digital holography is a technique that provides a non-invasive, label-free, quantitative, and high-resolution imaging employable in biological and science of matter fields, but not only. In the last decade, digital holography (DH) has undergone very significant signs of progress that made it one of the most powerful metrology tools. However, one of the most important issues to be afforded and solved for obtaining quantitative phase information about the analyzed specimen is related to phase aberrations. Sources of aberrations can be diverse, and several strategies have been developed and tested to make DH a reliable optical system with submicron resolution. This paper reviews the most effective and robust methods to remove or compensate phase aberrations in retrieved quantitative phase imaging by DH. Different strategies are presented and discussed in detail on how to remove or compensate for such disturbing aberrations. Among the various methods improvements in the optical setups are considered the numerical algorithms, the hybrid methods, and the very recent Artificial Intelligence (AI) approaches to compensate for all aberrations which affect the setups to improve the imaging quality and the accuracy of the reconstruction images’ procedures.