When COVID-19 first emerged, it was assumed that the virus induces only lung disease, but more and more studies have shown that the whole body gets affected.
Evidence shows that the brain and central nervous system (CNS) are affected by the virus. In a study of 214 COVID-19 patients 36% showed neurologic symptoms, such as CNS symptoms. Neurologic symptoms in patients with severe infections were even more common (45.5%). There is also evidence of patients without typical symptoms, such as fever, cough, anorexia and diarrhea, but neurologic manifestation. 
Another study found that meningitis/encephalitis is associated with SARS-CoV-2. In one patient, the virus was not detected in nasopharyngeal swab, but in cerebrospinal fluid. 
How exactly the virus affects the brain and nervous system is still unclear. Angiotensin-converting enzyme 2 (ACE2), the protein used by SARS-CoV-2 as a viral receptor for cell-entry, is expressed in both nervous system and skeletal muscles, suggesting direct infection by the virus . This theory is supported by the fact that traces of the virus were found in the cerebrospinal fluid . Other respiratory viruses enter the CNS through hematogenous or retrograde neuronal route. Since many COVID-19 patients experience smell impairment, this could also be the case for SARS-CoV-2.  However, secondary reasons, such as cytokine storm can also be potential causes for indirect damages in the brain and central nervous system.
Affects on the brain and CNS are most likely underestimated and more studies on this subject are necessary.
- Mao L, et al. Neurologic manifestations of hospitalized patients with coronavirus disease 2019 in Wuhan, China. JAMA neurology 77.6 (2020): 683-690.
- Moriguchi T, et al. A first case of meningitis/encephalitis associated with SARS-Coronavirus-2. International Journal of Infectious Diseases (2020).
- Hamming I, et al. Tissue distribution of ACE2 protein, the functional receptor for SARS coronavirus. A first step in understanding SARS pathogenesis. The Journal of Pathology: A Journal of the Pathological Society of Great Britain and Ireland 203.2 (2004): 631-637.