  We show that every quasiconformal contact foliation supports an invariant
metric and characterise such foliations by the dynamical property of
$C^1$-equicontinuity. We prove that a generalisation of the Weinstein
conjecture holds for quasiconformal contact foliations, and provide a lower
bound to the number of closed leaves. In particular, we show that the Weinstein
conjecture holds for quasiconformal Reeb fields.
