Moving habitat models are models about the dynamics of a population on a shifting habitat. These models are used to study climate-induced range shifts. In this talk, I will present some existing results from moving habitat models in the integrodifference equation framework, as well as some open problems that require further studies. These results and open problems include how population persistence depends on parameters such as the speed of shift, the diffusion rate, etc.; what dispersal strategies are evolutionarily stable in a moving habitat model; and how the population may lag behind the shifting habitat.