This talk contains two parts. I)We establish a large deviation principle for two-dimensional stochastic Navier-Stokes equations driven by multiplicative Levy noises. The weak convergence method introduced by Budhiraja, Dupuis and Maroulas [ Ann.Inst. Henri PoincaréProbab. Stat. 47 (2011) 725–747] plays a key role. This part is based on a joint work with Prof. Tusheng Zhang. II) We establish a moderate deviation principle for two-dimensional stochastic Navier-Stokes equations driven by multiplicative Levy noises. The weak convergence method introduced by Budhiraja, Dupuis and Ganguly in [Ann. Probab.
44 (2016), no. 3,1723–1775] plays a key role. This part is based on a joint work with Prof. Zhao Dong, Prof.Jie Xiong, Prof.Tusheng Zhang.