The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has spread around the globe causing coronavirus disease 2019 (COVID-19). Because it affects the respiratory system, common symptoms are cough and breathing difficulties with fever and fatigue. Also, some cases progress to acute respiratory distress syndrome (ARDS). The acute phase of COVID-19 has been also related to nervous system symptoms, including loss of taste and smell as well as encephalitis and cerebrovascular disorders. However, it remains unclear if neurological complications are due to the direct viral infection of the nervous system, or they appear as a consequence of the immune reaction against the virus in patients who presented pre-existing deficits or had a certain detrimental immune response. Importantly, the medium and long-term consequences of the infection by SARS-CoV-2 in the nervous system remain at present unknown. This review article aims to give an overview of the current neurological symptoms associated with COVID-19, as well as attempting to provide an insight beyond the acute affectation.