There are three weighted decompositions of tensors proposed in this paper, and the corresponding definitions of the weighted generalized tensor functions are given. The Cauchy integral formula of the weighted Moore-Penrose inverse is developed for solving the tensor equations. Besides above, we give the weighted projection tensors to discuss the representations of the weighted generalized power of tensors. Finally, some special tensors are studied which can preserve the structural invariance under the tensor functions defined in this paper.