  Isotropic Gaussian random fields on the sphere are characterized by
Karhunen-Lo\`{e}ve expansions with respect to the spherical harmonic functions
and the angular power spectrum. The smoothness of the covariance is connected
to the decay of the angular power spectrum and the relation to sample
H\"{o}lder continuity and sample differentiability of the random fields is
discussed. Rates of convergence of their finitely truncated Karhunen-Lo\`{e}ve
expansions in terms of the covariance spectrum are established, and algorithmic
aspects of fast sample generation via fast Fourier transforms on the sphere are
indicated. The relevance of the results on sample regularity for isotropic
Gaussian random fields and the corresponding lognormal random fields on the
sphere for several models from environmental sciences is indicated. Finally,
the stochastic heat equation on the sphere driven by additive, isotropic Wiener
noise is considered, and strong convergence rates for spectral discretizations
based on the spherical harmonic functions are proven.
