  We classify the local, polynomial, unitary gauge theories that violate
Lorentz symmetry explicitly at high energies and are renormalizable by weighted
power counting. We study the structure of such theories and prove that
renormalization does not generate higher time derivatives. We work out the
conditions to renormalize vertices that are usually non-renormalizable, such as
the two scalar-two fermion interactions and the four fermion interactions. A
number of four dimensional examples are presented.
