Although SARS-CoV-2 causes a respiratory viral infection, there is a large incidence of neurological complications occurring in COVID-19 patients. These range from headaches and loss of smell to encephalitis and strokes. Little is known about the likely diverse mechanisms causing these pathologies and there is a dire need to understand how to prevent and treat them. This review explores recent research from the perspective of investigating how the immune system could play a role in neurological complications, including cytokines, blood biomarkers, immune cells, and autoantibodies. We also discuss lessons learnt from animal models. Overall, we highlight two key points that have emerged from increasing evidence: (1) SARS-CoV-2 does not invade the brain in the majority of cases and so the associated neurological complications might arise from indirect effects, such as immune activation (2) although the immune system plays a critical role in controlling the virus, its dysregulation can cause pathology.