  A variety of lattice discretisations of continuum actions has been
considered, usually requiring the correct classical continuum limit. Here we
discuss "weird" lattice formulations without that property, namely lattice
actions that are invariant under most continuous deformations of the field
configuration, in one version even without any coupling constants. It turns out
that universality is powerful enough to still provide the correct quantum
continuum limit, despite the absence of a classical limit, or a perturbative
expansion. We demonstrate this for a set of O(N) models (or non-linear
$\sigma$-models). Amazingly, such "weird" lattice actions are not only in the
right universality class, but some of them even have practical benefits, in
particular an excellent scaling behaviour.
