The authors study distribution properties of continuous functions on compact connected homogeneous Riemannian manifolds \(X\) (generalizing known results in the special case \(X=\mathbb{R}^n/\mathbb{Z}^n)\). It is proved that almost all functions are uniformly distributed and almost no functions are well distributed. Similar results are obtained for sequences. The authors also announce a law of iterated logarithm for the discrepancy of functions on compact connected Riemannian manifolds [using results of \textit{W. Philipp}, Mem. Am. Math. Soc. 114 (1971; Zbl 0224.10052)].