The novel coronavirus severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the causative agent of coronavirus disease 2019 (COVID-19) and the ensuing worldwide pandemic. The spread of the virus has had global effects such as activity restriction, economic stagnation, and collapse of healthcare infrastructure. Severe SARS-CoV-2 infection induces a cytokine storm, leading to acute respiratory distress syndrome (ARDS) and multiple organ failure, which are very serious health conditions and must be mitigated or resolved as soon as possible. Mesenchymal stem cells (MSCs) and their exosomes can affect immune cells by inducing anti-inflammatory macrophages, regulatory T and B cells, and regulatory dendritic cells, and can inactivate T cells. Hence, they are potential candidate agents for treatment of severe cases of COVID-19. In this review, we report the background of severe cases of COVID-19, basic aspects and mechanisms of action of MSCs and their exosomes, and discuss basic and clinical studies based on MSCs and exosomes for influenza-induced ARDS. Finally, we report the potential of MSC and exosome therapy in severe cases of COVID-19 in recently initiated or planned clinical trials of MSCs (33 trials) and exosomes (1 trial) registered in 13 countries on ClinicalTrials.gov.