  A covering code is a set of codewords with the property that the union of
balls, suitably defined, around these codewords covers an entire space.
Generally, the goal is to find the covering code with the minimum size
codebook. While most prior work on covering codes has focused on the Hamming
metric, we consider the problem of designing covering codes defined in terms of
either insertions or deletions. First, we provide new sphere-covering lower
bounds on the minimum possible size of such codes. Then, we provide new
existential upper bounds on the size of optimal covering codes for a single
insertion or a single deletion that are tight up to a constant factor. Finally,
we derive improved upper bounds for covering codes using $R\geq 2$ insertions
or deletions. We prove that codes exist with density that is only a factor $O(R
\log R)$ larger than the lower bounds for all fixed~$R$. In particular, our
upper bounds have an optimal dependence on the word length, and we achieve
asymptotic density matching the best known bounds for Hamming distance covering
codes.
