  We study completely the Hilbert scheme and punctual Hilbert scheme of a nodal
curve, and the relative Hilbert scheme of a family of curves acquiring a node.
The results are then extended, less completely, to flag Hilbert schemes,
parametrizing chains of subschemes. We find, notably, that if the total space
$X$ of a family $X/B$ is smooth (over an algebraically closed field $\k$), then
the relative Hilbert scheme $Hilb_m(X/B)$ is smooth over $\k$ and the flag
Hilbert schemes are normal and locally complete intersection, but generally
singular.
