Shenfun is a high performance computing platform for solving partial differential equations (PDEs) by the spectral Galerkin method. The user interface to shenfun is very similar to FEniCS, but applications are limited to multidimensional tensor product grids, using either Cartesian or curvilinear grids (e.g., but not limited to, polar, cylindrical, spherical or parabolic). The code is parallelized with MPI through the mpi4py-fft package. Shenfun enables fast development of efficient and accurate PDE solvers (spectral order and accuracy), in the comfortable high-level Python language. The spectral accuracy is ensured by using high-order global orthogonal basis functions (Fourier, Legendre, Chebyshev, Laguerre, Hermite and Jacobi), as opposed to finite element codes that are using low-order local basis functions. Efficiency is ensured through vectorization (Numpy), parallelization (mpi4py) and by moving critical routines to Cython or Numba. Shenfun has been used to run turbulence simulations (Direct Numerical Simulations) on thousands of processors on high-performance supercomputers, see the spectralDNS repository.