  We simulate two-flavour lattice QCD with at a finite chemical potential
$\mu_I$ for isospin, and finite temperature. At small $\mu_I$, we determine the
position of the crossover from hadronic matter to a quark-gluon plasma as a
function of $\mu_I$. At larger $\mu_I$ we observe the phase transition from the
superfluid pion-condensed phase to a quark-gluon plasma, noting its change from
second order to first order as $\mu_I$ is increased. We also simulate
two-flavour lattice QCD at zero quark mass, using an action which includes an
additional 4-fermion interaction, at temperatures close to the chiral
transition on $N_t=8$ lattices.
