A newdefinition for circulant tensors is given, which is a generalization of the one for circulant matrices. Furthermore, we define the generalized circulant tensors which can be diagonalized by the Fourier matrix F and/ or F*. We also consider solving the circulant tensor systems by a fast algorithm based on the fast Fourier transform (FFT). Such algorithm is similar to that for circulant linear systems and it can also be performed in O(n log n).