In order to account for rotation and anelastic effects in the normal modes of the Earth, a structure of the space of normal modes different from those generally used in the elastic self-adjoint case is necessary. This can be done with a duality relation between the eigenproblem and the one obtained simply by reversing the Earth's rotation. This leads to new biorthonormality relations between modes and dual modes. Seismograms can then be expressed in terms of a normal mode expansion. The normal modes of an anelastic rotating Earth can be computed with perturbation theory. In order to take into account the coupling terms between different dispersion curves, as well as between toroidal and spheroidal modes, the perturbations start from an anelastic, nonrotating Earth rather than from an elastic one. The secular terms of the perturbation series, due to coupling effects between modes of the same multiplet, can then be removed. This ensures that higher order perturbation theory converges to the anelastic modes with sufficient accuracy, and gives, up to third order, expressions for the eigenmodes and eigenfrequencies. These expressions can be used to compute modes and siesmograms of an anelastic realistic Earth model, neglecting neither rotation, anelasticity, anisotropy or lateral heterogeneities. ¿American Geophysical Union 1991