We discuss the appealing properties of determinantal point process (DPP) models on the $d$-dimensional unit sphere $\mathbb S^d$, considering both the isotropic and the anisotropic case. DPPs are finite point processes exhibiting repulsiveness, but we also use them together with certain dependent thinnings when constructing point process models on $\mathbb S^d$ with aggregation on the large scale and regularity on the small scale. Moreover, for general point processes on $\mathbb S^d$, we present reduced Palm distributions and functional summary statistics, including nearest neighbour functions, empty space functions, and Ripley's and inhomogeneous $K$-functions. We conclude with a discussion on future work on statistics for spatial point processes on the sphere.

Keywords: aggregation; empty space function; inhomogeneous $K$-function; isotropic covariance function; joint intensities; likelihood; nearest neighbour function; Palm distribution; repulsiveness; spectral representation.