We derive fundamental asymptotic results for the expected covering radius $ρ(X_N)$ for $N$ points that are randomly and independently distributed with respect to surface measure on a sphere as well as on a class of smooth manifolds. For the unit sphere $\mathbb{S}^d \subset \mathbb{R}^{d+1}$, we obtain the precise asymptotic that $\mathbb{E}ρ(X_N)[N/\log N]^{1/d}$ has limit $[(d+1)\upsilon_{d+1}/\upsilon_d]^{1/d}$ as $N \to \infty $, where $\upsilon_d$ is the volume of the $d$-dimensional unit ball. This proves a recent conjecture of Brauchart et al. as well as extends a result previously known only for the circle. Likewise we obtain precise asymptotics for the expected covering radius of $N$ points randomly distributed on a $d$-dimensional ball, a $d$-dimensional cube, as well as on a 3-dimensional polyhedron (where the points are independently distributed with respect to volume measure). More generally, we deduce upper and lower bounds for the expected covering radius of $N$ points that are randomly and independently distributed on a metric measure space, provided the measure satisfies certain regularity assumptions.