A general theory to study the electromagnetic diffraction by imperfect half planes immersed in linear homogeneous bianisotropic media is presented. The problem is formulated in terms of Wiener-Hopf equations by deriving explicit spectral domain expressions for the characteristic impedances of bianisotropic media, which allow one to exploit their analytical properties. In the simpler case of perfect electric conducting and perfect magnetic conducting half planes, the Wiener-Hopf equations involve matrices of order 2, which can be factorized in closed form if the constitutive tensors of the bianisotropic material are of special form. Four of these special cases are discussed in detail. In order to deal with the more general problem, a technique to numerically factorize the Wiener-Hopf matrix kernels is presented. Our numerical approach is discussed on one example, by considering the previously unsolved problem of a perfect electric conducting half plane in a gyrotropic medium. The reported numerical results show that the diffracted field contribution is obtained by use of the saddle point integration method.