The current pandemic caused by the new severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has become a public health emergency. To date, March 1, 2021, coronavirus disease 2019 (COVID-19) has caused about 114 million accumulated cases and 2.53 million deaths worldwide. Previous pieces of evidence suggest that SARS-CoV-2 may affect the central nervous system (CNS) and cause neurological symptoms in COVID-19 patients. It is also known that angiotensin-converting enzyme-2 (ACE2), the primary receptor for SARS-CoV-2 infection, is expressed in different brain areas and cell types. Thus, it is hypothesized that infection by this virus could generate or exacerbate neuropathological alterations. However, the molecular mechanisms that link COVID-19 disease and nerve damage are unclear. In this review, we describe the routes of SARS-CoV-2 invasion into the central nervous system. We also analyze the neuropathologic mechanisms underlying this viral infection, and their potential relationship with the neurological manifestations described in patients with COVID-19, and the appearance or exacerbation of some neurodegenerative diseases.