In this article we consider polling systems with markovian server routings and where each station is attended according to a specific policy. A stationary regime for this system is constructed under general statistical assumptions (stationarity, ergodicity) on the input processes to the stations (in particular it is not required that these processes be mutually independent). The method of construction is as follows : one constructs recursively a sequence of stationary regimes for fictive systems that approximate in some sense the original polling system, the stationary regime is then identified as the limit of this sequence of stationary processes. The main tools for these results are Palm calculus and Birkhoff's ergodictheorem. It is shown by a coupling argument that this stationary regime is minimal in the stochastic ordering sense. The assumptions on the service policies allow to consider the purely gated policy, the a-limited policy, the binomial-gated policy and others. As a by-product sufficient conditions for the stationary regime of a G/G/1/0 queue with multiple server vacations are obtained.