The State of Cepheid Pulsation Theory

We review the current state of Cepheid modeling and discuss its dominant deficiency, namely the use of time dependent mixing length. Notwithstanding, Cepheid modeling has achieved some excellent successes, and we mention some of the most recent ones. Discrepancies between observations and modeling appear not so much in the gross properties of single mode Cepheids, but rather when more subtle nonlinear effects are important, such as in double mode or even triple mode pulsations. Finally we discuss what we consider the most important challenges for the next decade. These are, first, the multidimensional modeling of convection in a pulsating Cepheid environment, and second, the nonlinear modeling of the nonradial pulsations, and, possibly relatedly, of the Blazhko like phenomenon, both of which recently been observed in Cepheids.