  Isotropic positive definite functions on spheres play important roles in
spatial statistics, where they occur as the correlation functions of
homogeneous random fields and star-shaped random particles. In approximation
theory, strictly positive definite functions serve as radial basis functions
for interpolating scattered data on spherical domains. We review
characterizations of positive definite functions on spheres in terms of
Gegenbauer expansions and apply them to dimension walks, where monotonicity
properties of the Gegenbauer coefficients guarantee positive definiteness in
higher dimensions. Subject to a natural support condition, isotropic positive
definite functions on the Euclidean space $\mathbb{R}^3$, such as Askey's and
Wendland's functions, allow for the direct substitution of the Euclidean
distance by the great circle distance on a one-, two- or three-dimensional
sphere, as opposed to the traditional approach, where the distances are
transformed into each other. Completely monotone functions are positive
definite on spheres of any dimension and provide rich parametric classes of
such functions, including members of the powered exponential, Mat\'{e}rn,
generalized Cauchy and Dagum families. The sine power family permits a
continuous parameterization of the roughness of the sample paths of a Gaussian
process. A collection of research problems provides challenges for future work
in mathematical analysis, probability theory and spatial statistics.
