  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.
