The paper provides an introduction to the concept of Probabilistic Safety Assessment, an evaluation of its recent developments, and perspectives on the future research directions in this area. To do so, a conceptual understanding to safety assessment is first provided, followed by an introduction to what Probabilistic Safety Assessment is about. From this, the historical background and development of Probabilistic Safety Assessment in the context of nuclear safety are discussed, including a brief description and evaluation of some methods implemented to perform such analysis. After this, the paper reviews some of the recent research developments in Probabilistic Safety Assessment in the aspects of multi-unit safety assessment, dynamic Probabilistic Safety Assessment, reliability analysis, cyber-security, and policy-making. Each aspect is elaborated in detail, with perspectives provided on its potential limitations. Finally, the paper discusses research topics in six areas and challenges within the Probabilistic Safety Assessment discipline, for which further investigation might be conducted in the future. Hence, the objectives of the review paper are (1) to serve as a tutorial for readers who are new to the concept of Probabilistic Safety Assessment; (2) to provide a historical perspective on the development of the Probabilistic Safety Assessment field over the past seven decades; (3) to review the state-of-the-art developments in the use of Probabilistic Safety Assessment in the context of nuclear safety; (4) to provide an evaluative perspective on the methods implemented for Probabilistic Safety Assessment within the current literature; and (5) to provide perspectives on the future research directions that can potentially be explored, thereby also targeting the wider research community within the nuclear safety discipline towards pushing the frontiers of Probabilistic Safety Assessment research.