  We study a class of perturbative scalar quantum field theories where dynamics
is characterized by Lorentz-invariant or Lorentz-breaking non-local operators
of fractional order and the underlying spacetime has a varying spectral
dimension. These theories are either ghost free or power-counting
renormalizable but they cannot be both at the same time. However, some of them
are one-loop unitary and finite, and possibly unitary and finite at all orders.
