This paper addresses the dynamic assignment of a given set of surgeries to multiple identical operating rooms (ORs). Surgeries have random durations and planned surgeon arrival times. Surgeries are assigned dynamically to ORs at surgery completion events. The goal is to minimize the total expected cost incurred by surgeon waiting, OR idling, and OR overtime. We first formulate the problem as a multi-stage stochastic programming model. An efficient algorithm is then proposed by combining a two-stage stochastic programming approximation and some look-ahead strategies. A perfect information-based lower bound of the optimal expected cost is given to evaluate the optimality gap of the dynamic assignment strategy. Numerical results show that the dynamic scheduling and optimization with the proposed approach significantly improve the performance of static scheduling and First Come First Serve (FCFS) strategy.