This thesis attempts to capture recent developments related to numerical integration on spheres of arbitrary dimension. More precisely, the quality of quasi-Monte Carlo rules on Bessel potential spaces is studied in an Information-based Complexity setting via a worst-case error criterion. Further, uniform distribution of sequences of point sets is analyzed by means of spherical cap discrepancy, and connections to cubature errors are presented. Combinatorial methods are employed to obtain non-constructive upper bounds on the minimal cap discrepancy and candidates for low-discrepancy sequences on the sphere are examined, both theoretically and numerically. Additionally, a framework built on minimal energy configurations is introduced, in which geometric kernels generating reproducing kernel Hilbert spaces on the sphere are embedded. Via Stolarsky's principle, lower bounds for worst-case errors in these spaces are used to derive a lower bound on cap discrepancy. Generalizations of this relation were shown to give explicit formulas for computing worst-case errors in potential spaces of higher smoothness.
