In this thesis the author's work on the computational modelling and the experimental measurement of millimetre and sub-millimetre wave horn antennas for Cosmic Microwave Background (CMB) experiments is presented. This computational work particularly concerns the analysis of the multimode channels of the High Frequency Instrument (HFI) of the European Space Agency (ESA) Planck satellite using mode matching techniques to model their farfield beam patterns. To undertake this analysis the existing in-house software was upgraded to address issues associated with the stability of the simulations and to introduce additional functionality through the application of Single Value Decomposition in order to recover the true hybrid eigenfields for complex corrugated waveguide and horn structures. The farfield beam patterns of the two highest frequency channels of HFI (857 GHz and 545 GHz) were computed at a large number of spot frequencies across their operational bands in order to extract the broadband beams. The attributes of the multimode nature of these channels are discussed including the number of propagating modes as a function of frequency. A detailed analysis of the possible effects of manufacturing tolerances of the long corrugated triple horn structures on the farfield beam patterns of the 857 GHz horn antennas is described in the context of the higher than expected sidelobe levels detected in some of the 857 GHz channels during flight. Additionally the pre-flight measurements of the flight horns and qualification horn are analysed in detail which verifies the multimode nature of the horns. This computational work is complemented by a novel approach to the measurement of millimetre-wave antennas using digital holographic techniques particularly with the location of their phase centres in mind. The measurement at 100 GHz of a horn antenna specially designed for future CMB polarisation experiments is presented. Finally some additional applications of millimetre-wave holography are discussed.