  We implement the theory of resonant scattering in the context of Cosmic
Microwave Background (CMB) polarisation anisotropies. We compute the changes in
the E-mode polarisation (EE) and Temperature E-mode (TE) CMB power spectra
introduced by the scattering on a resonant transition with a given optical
depth $\tau_X$ and polarisation coefficient $E_1$. The latter parameter,
accounting for how anisotropic the scattering is, depends on the exchange of
angular momentum in the transition and enables, {\it a priori}, to distinguish
among different possible resonances. We use this formalism in two different
scenarios: cosmological recombination and cosmological reionisation. In the
context of cosmological recombination, we compute predictions in frequency and
multipole space for the change in the TE and EE power spectra introduced by
scattering on the $H_{\alpha}$ and $P_{\alpha}$ lines of Hydrogen. This
constitutes a fundamental test for the standard model of recombination, and the
sensitivity it requires is comparable to that needed in measuring the
primordial CMB B-mode polarisation component. In the context of reionisation,
we study the scattering off metals and ions produced by the first stars, and
find that polarisation anisotropies, apart from providing a consistency test
for intensity measurements, give some insight on how reionisation evolved:
since they keep memory of how anisotropic the line scattering is, they should
be able to discern the OI 63.2$\mu$m transition from other possible transitions
associated to OIII, NII, NIII, etc. The amplitude of these signals are,
however, between 10 to 100 times below the (already challenging) level of CMB
B-mode polarisation anisotropies.
