We present a detailed analysis of present and future cosmic microwave background (CMB) constraints of the value of the fine-structure constant, α. We carry out a more detailed analysis of the WMAP first-year data, deriving state-of-the-art constraints on α and discussing various other issues, such as the possible hints for the running of the spectral index. We find, at 95 per cent confidence level, that 0.95 < α_dec/α₀ < 1.02. Setting dn_S/d ln k= 0 yields 0.94 < α_dec/α₀ < 1.01 as previously reported. We find that a lower value of α_dec/α₀ makes a value of dn_S/d ln k= 0 more compatible with the data. We also perform a thorough Fisher matrix analysis (including both temperature and polarization, as well as α and the optical depth τ), in order to estimate how future CMB experiments will be able to constrain α and other cosmological parameters. We find that Planck data alone can constrain τ with an accuracy of the order 4 per cent and that this constraint can be as small as 1.7 per cent for an ideal cosmic variance limited (CVL) experiment. Constraints on α are of the order of 0.3 per cent for Planck and can in principle be as small as 0.1 per cent using CMB data alone: tighter constraints will require further (non-CMB) priors.