Heat kernel coefficients on the sphere in any dimension

We derive all heat kernel coefficients for Laplacians acting on scalars, vectors, and tensors on fully symmetric spaces, in any dimension. Final expressions are easy to evaluate and implement, and confirmed independently using spectral sums and the Euler-Maclaurin formula. We also obtain the Green's function for Laplacians acting on transverse traceless tensors in any dimension, and new integral representations for heat kernels using known eigenvalue spectra of Laplacians. Applications to quantum gravity and the functional renormalisation group, and other, are indicated.