Chebfun2 is an extension of Chebfun to 2D scalar- and vector-valued functions defined on rectangles. The main approximation scheme relies on a iterative variant of Gaussian elimination on functions to compute low rank approximants. Related low rank ideas are now employed on partial differential operators, which has allowed us to extend a well-conditioned spectral method for ODEs to a fast and spectrally accurate PDE solver. Some PDEs requiring over a million degrees of freedom can be now be solved in under a second. Software is publicly available as part of Chebfun.