  We study a class of Lorentz violating quantum field theories that contain
higher space derivatives, but no higher time derivatives, and become
renormalizable in the large N expansion. The fixed points of their
renormalization-group flows provide examples of exactly "weighted scale
invariant" theories, which are noticeable Lorentz violating generalizations of
conformal field theories. We classify the scalar and fermion models that are
causal, stable and unitary. Solutions exist also in four and higher dimensions,
even and odd. In some explicit four dimensional examples, we compute the
correlation functions to the leading order in 1/N and the critical exponents to
the subleading order. We construct also RG flows interpolating between pairs of
fixed points.
