This paper deals with the stability of periodic polling models with mixed service policies. The interarrivals to all queues are independent and exponentially distributed and the service and the switch-over times are independent with general distributions. The necessary and sufficient condition for the stability of such polling systems is established. The proof is based on the stochastic monotonicity of the state process at the polling instants. The stability of only a subset of the queues is also analyzed, and, in case of heavy traffic, the order of explosion of the queues is given.