This thesis studies characterisations and properties of spatial and spatio-temporal Gaussian processes defined over the sphere (or in the spatio-temporal case the product of the sphere and the real line). Such processes are of importance in global weather and climate science, where the geometry is necessarily spherical, but, especially in the dynamic setting, they are less well-studied than their Euclidean counterparts.

Beginning with Brownian motion, we first look at characterising Gaussian randomness on the sphere and sphere-cross-line, and how it compares with the Euclidean setting -- we show that the characterisation theorems of Gaussian processes on spaces of types spanning the real line, the sphere and sphere-cross-line can be phrased as consequences of a powerful general theorem of harmonic analysis. We go on to find the answer to a recent question posed about dimension-hopping operators for positive-definite (i.e. covariance) functions on the sphere-cross-line, and consider how we could go about constructing dimension-hopping operators with the semi-group property on the sphere.

Later, we address the theory of the path properties of these processes, extending a finite-dimensional result the the infinite-dimensional case and showing that a remarkably elegant approach for processes on Euclidean space carries over to our setting. We finish by finding the analogue of the powerful Ciesielski isomorphism for continuous functions on the two-sphere.