Aortic aneurysm is a cardiovascular disease related to the alteration of the aortic tissue. It is an important cause of death in developed countries, especially for older patients. The diagnosis and treatment of such pathology is performed according to guidelines, which suggest surgical or interventional (stenting) procedures for aneurysms with a maximum diameter above a critical threshold. Although conservative, this clinical approach is also not able to predict the risk of acute complications for every patient. In the last decade, there has been growing interest towards the development of advanced in silico aortic models, which may assist in clinical diagnosis, surgical procedure planning or the design and validation of medical devices. This paper details a comprehensive review of computational modelling and simulations of blood vessel interaction in aortic aneurysms and dissection, following the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA). In particular, the following questions are addressed: “What mathematical models were applied to simulate the biomechanical behaviour of healthy and diseased aortas?” and “Why are these models not clinically implemented?”. Contemporary evidence proves that computational models are able to provide clinicians with additional, otherwise unavailable in vivo data and potentially identify patients who may benefit from earlier treatment. Notwithstanding the above, these tools are still not widely implemented, primarily due to low accuracy, an extensive reporting time and lack of numerical validation.