Field theories with spontaneous Lorentz violation involving an antisymmetric 2-tensor are studied. A general action including nonminimal gravitational couplings is constructed, and features of the Nambu- Goldstone and massive modes are discussed. Minimal models in Minkowski spacetime exhibit dualities with Lorentz-violating vector and scalar theories. The post-Newtonian expansion for nonminimal models in Riemann spacetime involves qualitatively new features, including the absence of an isotropic limit. Certain interactions producing stable Lorentz-violating theories in Minkowski spacetime solve the renormalization-group equations in the tadpole approximation.