© 2015, Biomedical Engineering Society. Dissection of an artery is characterised by the separation of the layers of the arterial wall causing blood to flow within the wall. The incidence rates of thoracic aortic dissection (AoD) are increasing, despite falls in virtually all other manifestations of cardiovascular disease, including abdominal aortic aneurysm (AAA). Dissections involving the ascending aorta (Type A) are a medical emergency and require urgent surgical repair. However, dissections of the descending aorta (Type B) are less lethal and require different clinical management whereby the patient may not be offered surgery unless complicating factors are present. But how do we tell if a patient will develop a complication later on? Currently, there is no consensus and the evidence base is limited. There is an opportunity for computational biomechanics to help clinicians decide as to which cases to repair and which to manage with blood pressure control. In this review article, we look at AoD from both the clinical and biomechanical perspective and discuss some of the recent computational studies of both Type A and B AoD. We then focus more on Type B where the real opportunity for patient-specific modelling exists. Finally, we look ahead at some of the promising areas of research that may help clinicians improve the decision-making process surrounding Type B AoD.