The SARS-CoV-2 virus, first reported in December 2019 in China, is the causative agent of the current COVID-19 pandemic that, at the time of writing (1 November 2020) has infected almost 43 million people and caused the death of more than 1 million people. The spectrum of clinical manifestations observed during COVID-19 infection varies from asymptomatic to critical life-threatening clinical conditions. Emerging evidence shows that COVID-19 affects far more organs than just the respiratory system, including the heart, kidneys, blood vessels, liver, as well as the central nervous system (CNS) and the peripheral nervous system (PNS). It is also becoming clear that the neurological and psychological disturbances that occur during the acute phase of the infection may persist well beyond the recovery. The aim of this review is to propel further this emerging and relevant field of research related to the pathophysiology of neurological manifestation of COVID-19 infection (Neuro-COVID). We will summarize the PNS and CNS symptoms experienced by people with COVID-19 both during infection and in the recovery phase. Diagnostic and pharmacological findings in this field of study are strongly warranted to address the neurological and psychological symptoms of COVID-19.