The new coronavirus SARS-CoV-2 and the disease it causes COVID-19 involves not only respiratory system damage, but can also lead to disorders of the central and peripheral nervous system, as well as the muscular system. This article presents published data and our own observations on the course of neurological disorders in COVID-19 patients. There is a relationship between the severity of COVID-19 and the severity and frequency of neurological manifestations. Severe neurological disorders are mostly seen in severe cases of COVID-19 and include acute cerebrovascular accidents (aCVA), acute necrotizing encephalopathy, and Guillain–Barré syndrome. Factors potentially complicating the course of COVID-19 and increasing the development of neurological complications include arterial hypertension, diabetes mellitus, and chronic cardiac and respiratory system diseases. Questions of the possible effects of human coronaviruses on the course of chronic progressive neurological diseases are addressed using multiple sclerosis (MS) as an example. We discuss the management of patients with aCVA and MS depending on the risk of developing coronavirus infection.