  We give a complete answer to the following basic question: "What is the
maximal fraction of deletions or insertions tolerable by $q$-ary list-decodable
codes with non-vanishing information rate?"
  This question has been open even for binary codes, including the restriction
to the binary insertion-only setting, where the best-known result was that a
$\gamma\leq 0.707$ fraction of insertions is tolerable by some binary code
family.
  For any desired $\epsilon > 0$, we construct a family of binary codes of
positive rate which can be efficiently list-decoded from any combination of
$\gamma$ fraction of insertions and $\delta$ fraction of deletions as long as $
\gamma+2\delta\leq 1-\epsilon$. On the other hand, for any $\gamma,\delta$ with
$\gamma+2\delta=1$ list-decoding is impossible. Our result thus precisely
characterizes the feasibility region of binary list-decodable codes for
insertions and deletions.
  We further generalize our result to codes over any finite alphabet of size
$q$. Surprisingly, our work reveals that the feasibility region for $q>2$ is
not the natural generalization of the binary bound above. We provide tight
upper and lower bounds that precisely pin down the feasibility region, which
turns out to have a $(q-1)$-piece-wise linear boundary whose $q$ corner-points
lie on a quadratic curve.
  The main technical work in our results is proving the existence of code
families of sufficiently large size with good list-decoding properties for any
combination of $\delta,\gamma$ within the claimed feasibility region. We
achieve this via an intricate analysis of codes introduced by [Bukh, Ma; SIAM
J. Discrete Math; 2014]. Finally, we give a simple yet powerful concatenation
scheme for list-decodable insertion-deletion codes which transforms any such
(non-efficient) code family (with vanishing information rate) into an
efficiently decodable code family with constant rate.
