  We develop a topological theory for disordered Weyl semimetals in the
framework of gauge invariance of replica formalism and boundary-bulk
correspondence of Chern insulators. An anisotropic topological $\theta$-term is
analytically derived for the effective non-linear sigma model. It is this
nontrivial topological term that ensures the bulk transverse transport of Weyl
semimetals to be robust against disorders. Moreover, we establish a general
diagram that reveals the intrinsic relations among topological terms in the
non-linear sigma models and gauge response theories respectively for $(2n +
2)$-dimensional topological insulators, $(2n+1)$-dimensional chiral fermions,
$(2n+1)$-dimensional chiral semimetals, and $(2n)$-dimensional topological
insulators with $n$ being a positive integer.
