A myriad of factors have contributed to urban congestion in cities around the world where millions of hours are wasted on the road. Such problems are likely to exacerbate as the population density is expected to outpace gains in capacity in a majority of cities. We introduce an innovation to radically improve urban mobility which is made possible by concurrent advancements in aircraft vehicle efficiency, battery technology, air traffic control, noise, emissions, and ridesharing. The world where one can push a button and get a flight is closer than most realize, although significant operational and logistical challenges exist due to uncertainties in demand and future technologies. In addition to providing an overview of urban on-demand aviation, we will survey how optimization and machine learning are being used to inform the design and resource allocation of this promising new mode of transportation.