Coronavirus Disease 2019 is predominantly a disorder of the respiratory system, but neurological complications have been recognised since early in the pandemic. The major pathophysiological processes leading to neurological damage in COVID-19 are cerebrovascular disease, immunologically mediated neurological disorders and the detrimental effects of critical illness on the nervous system. It is still unclear whether direct invasion of the nervous system by the Severe Acute Respiratory Syndrome Coronavirus 2 occurs; given the vast numbers of people infected at this point, this uncertainty suggests that nervous system infection is unlikely to represent a significant issue if it occurs at all. In this review, we explore what has been learnt about the neurological complications of COVID-19 over the course of the pandemic, and by which mechanisms these complications most commonly occur.