  Let $V$ be an $n$-set, and let $X$ be a random variable taking values in the
powerset of $V$. Suppose we are given a sequence of random coupons $X_1, X_2,
\ldots $, where the $X_i$ are independent random variables with distribution
given by $X$. The covering time $T$ is the smallest integer $t\geq 0$ such that
$\bigcup_{i=1}^tX_i=V$. The distribution of $T$ is important in many
applications in combinatorial probability, and has been extensively studied.
However the literature has focussed almost exclusively on the case where $X$ is
assumed to be symmetric and/or uniform in some way.
  In this paper we study the covering time for much more general random
variables $X$; we give general criteria for $T$ being sharply concentrated
around its mean, precise tools to estimate that mean, as well as examples where
$T$ fails to be concentrated and when structural properties in the distribution
of $X$ allow for a very different behaviour of $T$ relative to the
symmetric/uniform case.
