Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the etiologic agent of COVID-19, a disease that as of July 10, 2020, has infected >12 million people and killed >500,000. COVID-19 infection leads to acute respiratory distress syndrome in a subset of patients and is a primary driver of acute morbidity in infected persons. However, it is becoming increasingly clear that SARS-CoV-2 infection drives dysfunction and pathology outside the lungs, including reports of renal, cardiac, and neurological complications. In this study, we summarize the known incidence and evidence of neurological complications associated with SARS-CoV-2 infection and other pathogenic coronaviruses. These studies describe a poorly understood spectrum of COVID-19 central nervous system symptoms, ranging from common and subclinical issues such as anosmia and headache to more concerning reports of stroke and encephalopathy. We discuss potential mechanisms of pathogenesis, including a discussion of how the understanding of neurological complications known to occur in HIV-1 patients may provide insight into SARS-CoV-2-associated neurological manifestations. Specifically, three hypotheses are discussed that are informed by decades of knowledge about HIV pathogenesis in the brain, which include a potential direct viral effect, an indirect viral effect, and/or a neuroimmune axis effect. Individually or in combination these potential effects may contribute to COVID-19 neurological complications.