In this talk, we first consider a fractional chemotaxis system coupled with the Navier-Stokes fluid in the whole space R^N for N≥3. With the help of selecting appropriate functional space, we develop a framework for a unified treatment of the existence, uniqueness and decay estimates of global mild solution to this problem under the assumption that initial data are small enough. Then we deal with a coupled chemotaxis-Navier-Stokes system with logistic source and a fractional diffusion on three-dimensional periodic torus T^3, our main purpose is to investigate the global existence of weak solutions to this system in the case of a weaker diffusion, and after some waiting time, the weak solutions in fact become smooth and converge to a semi-trivial steady state.