In recent months it has emerged that the novel coronavirus—responsible for the COVID-19 pandemic—causes reduction of smell and taste in a large fraction of patients. The chemosensory deficits are often the earliest, and sometimes the only signs in otherwise asymptomatic carriers of the SARS-CoV-2 virus. The reasons for the surprisingly early and specific chemosensory dysfunction in COVID-19 are now beginning to be elucidated. In this hypothesis review, we discuss implications of the recent finding that the prevalence of smell and taste dysfunction in COVID-19 patients differs between populations, possibly because of differences in the spike protein of different virus strains or because of differences in the host proteins that enable virus entry, thus modifying infectivity. We review recent progress in defining underlying cellular and molecular mechanisms of the virus-induced anosmia, with a focus on the emerging crucial role of sustentacular cells in the olfactory epithelium. We critically examine the current evidence whether and how the SARS-CoV-2 virus can follow a route from the olfactory epithelium in the nose to the brain to achieve brain infection, and we discuss the prospects for using the smell and taste dysfunctions seen in COVID-19 as an early and rapid diagnostic screening tool.