In cities around the world, transit is currently provided with fixed route transportation only, whence the inherent limited Quality of Service (QoS) for travelers in sub-urban areas and during off-peak. On the other hand, it has been shown that completely replacing fixed-route with demand-responsive transit fails to serve the high transportation demand during peak hours. Therefore, it is still unclear how we can maximize the potential of demand-responsive transit by satisfying the complicated demand pattern varying with time and space. In this paper we propose Flexible Transit, a transit system design which gets the best from fixed-route and demand-responsive transit, depending on the demand observed in each sub-region of the urban conurbation and time-of-day. The goal is to provide high transportation capacity while guaranteeing high QoS, two objectives that are instead conflicting with classic fixed-schedule transportation. To this end, we first resort to microsimulation to show the limits of using either only fixed-route buses or only demand-responsive buses. This motivates the need of alternating between them instead, which we do in Flexible Transit. We then resort to Continuous Approximation to find the optimal design of flexible transit. We show that the flexible transit can significantly improve the user costs, in particular in suburban areas, while also reducing the overall cost of user and operator. We believe our findings suggest important policy insights in designing and planning of future transit systems, to take full advantage of demand-adaptive transportation modes.