Perfect Actions for Scalar Theories

We construct an optimally local perfect lattice action for free scalars of arbitrary mass, and truncate its couplings to a unit hypercube. Spectral and thermodynamic properties of this ``hypercube scalar’’ are drastically improved compared to the standard action. We also discuss new variants of perfect actions, using anisotropic or triangular lattices, or applying new types of RGTs. Finally we add a \lambda \phi^4 term and address perfect lattice perturbation theory. We report on a lattice action for the anharmonic oscillator, which is perfect to O(\lambda).