In this paper, a fast finite volume method is proposed for the initial and boundary value problems of spatial fractional diffusion equations on nonuniform meshes. The discretizations of the Riemann-Liouville fractional derivatives lead to unstructured dense coefficient matrices, differing from the Toeplitz-like structure under the uniform mesh. The fast algorithm is proposed by using the sum-of-exponentials (SOE) technique to the spatial kernel x,α∈(0,1). Then, the matrix-vector multiplications of the resulting coefficient matrices could be implemented in O(mlog⁡m) operations, where m denotes the size of matrices. Iterative solvers are preferably applied to obtain the numerical solution. The proposed fast scheme is proved to be unconditionally stable for sufficiently accurate SOE approximation. Meanwhile, a banded preconditioner is exploited to accelerate the Krylov subspace method. Numerical experiments are provided to demonstrate the efficiency of the proposed fast algorithm.