Peer-to-Peer (P2P) technology has recently been employed to deliver large-scale video multicast services at low server infrastructure cost. A number of commercial systems have been successfully deployed on the Internet to stream live and on-demand videos to millions of users. In order to gain insights into the success of P2P video streaming, we have undertaken extensive measurement and analysis to investigate how loosely coordinated individual peer behaviors collectively converge to predictable system-wide video streaming quality. In the first part of this talk, I will share with you our recent measurement results on a popular P2P streaming system, namely PPLive, with a focus on the phenomena of “Large Numbers”. The second part of this talk is devoted to exposing the “Law of Large Numbers” for generic P2P streaming systems. We establish the fundamental performance bounds, including the maximum supportable streaming rate and minimum achievable streaming delay, and develop practical P2P streaming designs to achieve the derived bounds in realistic P2P network environment.