Abstract: Nearby cycles functor was defined by Deligne to study Milnor fiber of holomorphic functions. While motivic reincarnation of the Milnor fiber was defined by Denef-Loeser, called motivic Milnor fiber. They bring a lot of information of the Milnor fiber such as Euler characteristic, monodromy, Hodge spectrum.  In case of meromorphic functions, they define some locally trivial bundle over small neighbourhood of indeterminant point and its fiber is called Milnor fiber. In this talk, we  introduce meromorphic nearby cycle functor and motivic Milnor fiber  for rational functions. As application, we use them to study monodromies of meromorphic functions in various situations (this is join work with Kiyoshi Takeuchi).