The stability of polling models with several servers and Markovian routing of the servers is analyzed. Customers arrive according to independent renewal processes. The service times and the switch-over times are independent with general distributions, independent of the servers. Service policies are attached to the queues. We established the necessary and sufficient condition for stability for generalservice policies covering the classical ones. The sufficiency of the condition is proved using the associated fluid model. For the necessity of the condition,a straightforward argument is given. Moreover, local stability is given.