  The last decades have seen an unprecedented increase in the availability of
data sets that are inherently global and temporally evolving, from remotely
sensed networks to climate model ensembles. This paper provides a view of
statistical modeling techniques for space-time processes, where space is the
sphere representing our planet. In particular, we make a distintion between (a)
second order-based, and (b) practical approaches to model temporally evolving
global processes. The former are based on the specification of a class of
space-time covariance functions, with space being the two-dimensional sphere.
The latter are based on explicit description of the dynamics of the space-time
process, i.e., by specifying its evolution as a function of its past history
with added spatially dependent noise.
  We especially focus on approach (a), where the literature has been sparse. We
provide new models of space-time covariance functions for random fields defined
on spheres cross time. Practical approaches, (b), are also discussed, with
special emphasis on models built directly on the sphere, without projecting the
spherical coordinate on the plane.
  We present a case study focused on the analysis of air pollution from the
2015 wildfires in Equatorial Asia, an event which was classified as the year's
worst environmental disaster. The paper finishes with a list of the main
theoretical and applied research problems in the area, where we expect the
statistical community to engage over the next decade.
