  The Reconstruction Conjecture of Ulam asserts that, for $n\geq 3$, every
$n$-vertex graph is determined by the multiset of its induced subgraphs with
$n-1$ vertices. The conjecture is known to hold for various special classes of
graphs but remains wide open. We survey results on the more general conjecture
by Kelly from 1957 that for every positive integer $\ell$ there exists $M_\ell$
(with $M_1=3$) such that when $n\geq M_\ell$ every $n$-vertex graph is
determined by the multiset of its induced subgraphs with $n-\ell$ vertices.
