  Many extremal problems for graphs have threshold graphs as their extremal
examples. For instance the current authors proved that for fixed $k\ge 1$,
among all graphs on $n$ vertices with $m$ edges, some threshold graph has the
fewest matchings of size $k$; indeed either the lex graph or the colex graph is
such an extremal example. In this paper we consider the problem of maximizing
the number of matchings in the class of threshold graphs. We prove that the
minimizers are what we call \emph{almost alternating threshold graphs}.
  We also discuss a problem with a similar flavor: which threshold graph has
the fewest independent sets. Here we are inspired by the result that among all
graphs on $n$ vertices and $m$ edges the lex graph has the most independent
sets.
