Adaptive streaming, such as Dynamic Adaptive Streaming over HTTP (DASH), has been widely deployed to provide uninterrupted video streaming service to users with dynamic network conditions. In this paper, we analytically study the potential of using P2P in conjunction with adaptive streaming. We first study the capacity of P2P adaptive streaming by developing utility maximization models that take into account peer heterogeneity, taxation-based incentives, multi-version videos at discrete rates. We further develop stochastic models to study the performance of P2P adaptive streaming in face of bandwidth variations and peer churn. Through analysis and simulations, we demonstrate that incentive-compatible video sharing between peers can be easily achieved with simple video coding and distribution designs. P2P adaptive streaming not only significantly reduces the load on the servers, but also improves the stability of user-perceived video quality in the face of dynamic bandwidth changes.