  We construct nonlinear extensions of Dirac's relativistic electron equation
that preserve its other desirable properties such as locality, separability,
conservation of probability and Poincar\'e invariance. We determine the
constraints that the nonlinear term must obey and classify the resultant
non-polynomial nonlinearities in a double expansion in the degree of
nonlinearity and number of derivatives. We give explicit examples of such
nonlinear equations, studying their discrete symmetries and other properties.
Motivated by some previously suggested applications we then consider nonlinear
terms that simultaneously violate Lorentz covariance and again study various
explicit examples. We contrast our equations and construction procedure with
others in the literature and also show that our equations are not gauge
equivalent to the linear Dirac equation. Finally we outline various physical
applications for these equations.
