An emerging new mission for aeronautics is Urban Air Mobility (UAM), a concept for air transportation around metropolitan areas with passenger-carrying operations. UAM vehicles must be capable of vertical take-off and landing, and this requirement presents unique technical challenges for electric and hybrid-based vertical take-off and landing (eVTOL). A critical challenge for UAM market growth is to gain public acceptance for being as safe as - or safer than - commercial air travel and automotive transportation. There is a lack of data for propulsion systems, components, and the associated thermal management systems for UAM eVTOL propulsion systems. The new mission, new propulsion system concepts, safety criticality of propulsion component performance during vertical take-off and lift operations, and lack of data presents many research challenges and opportunities. NASA has developed and published UAM vehicle concept studies. For a subset of the said concept vehicles, NASA has contracted for a study to identify failure modes and hazards associated with the propulsion systems of the concept vehicles and to perform functional hazard analyses (FHA) and failure modes and effects criticality analyses (FMECA) for each. From the completed study results, it was recommended for NASA to support research toward developing electric/hybrid-electric propulsion components with improved reliability and to explore powertrain architectures that can take advantage of higher reliability components to achieve inherent air-vehicle safety. NASA has started a research effort for UAM propulsion with a focus toward improving safety and reliability. Recent results and research strategy will be discussed toward the goals by means of: 1) improving individual component reliability through advanced materials and design methods, 2) improving the thermal management system, and 3) designing propulsion system architectures to provide inherent UAM vehicle safety.