A comprehensive review of the neurological disorders reported during the current COVID-19 pandemic demonstrates that infection with SARS-CoV-2 affects the central nervous system (CNS), the peripheral nervous system (PNS) and the muscle. CNS manifestations include: headache and decreased responsiveness considered initial indicators of potential neurological involvement; anosmia, hyposmia, hypogeusia, and dysgeusia are frequent early symptoms of coronavirus infection. Respiratory failure, the lethal manifestation of COVID-19, responsible for 264,679 deaths worldwide, is probably neurogenic in origin and may result from the viral invasion of cranial nerve I, progressing into rhinencephalon and brainstem respiratory centers. Cerebrovascular disease, in particular large-vessel ischemic strokes, and less frequently cerebral venous thrombosis, intracerebral hemorrhage and subarachnoid hemorrhage, usually occur as part of a thrombotic state induced by viral attachment to ACE2 receptors in endothelium causing widespread endotheliitis, coagulopathy, arterial and venous thromboses. Acute hemorrhagic necrotizing encephalopathy is associated to the cytokine storm. A frontal hypoperfusion syndrome has been identified. There are isolated reports of seizures, encephalopathy, meningitis, encephalitis, and myelitis. The neurological diseases affecting the PNS and muscle in COVID-19 are less frequent and include Guillain-Barré syndrome; Miller Fisher syndrome; polyneuritis cranialis; and rare instances of viral myopathy with rhabdomyolysis. The main conclusion of this review is the pressing need to define the neurology of COVID-19, its frequency, manifestations, neuropathology and pathogenesis. On behalf of the World Federation of Neurology we invite national and regional neurological associations to create local databases to report cases with neurological manifestations observed during the on-going pandemic. International neuroepidemiological collaboration may help define the natural history of this worldwide problem.