  We derive fundamental asymptotic results for the expected covering radius
$\rho(X_N)$ 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 $\mathbb{E}\rho(X_N)[N/\log N]^{1/d}$ 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.
