  We study point processes on $\mathbb S^d$, the $d$-dimensional unit sphere
$\mathbb S^d$, considering both the isotropic and the anisotropic case, and
focusing mostly on the spherical case $d=2$. The first part studies reduced
Palm distributions and functional summary statistics, including nearest
neighbour functions, empty space functions, and Ripley's and inhomogeneous
$K$-functions. The second part partly discusses the appealing properties of
determinantal point process (DPP) models on the sphere and partly considers the
application of functional summary statistics to DPPs. In fact DPPs exhibit
repulsiveness, but we also use them together with certain dependent thinnings
when constructing point process models on the sphere with aggregation on the
large scale and regularity on the small scale. We conclude with a discussion on
future work on statistics for spatial point processes on the sphere.
