Photonic signal processing offers the advantages of large time-bandwidth capabilities to overcome inherent electronic limitations. In-fibre signal processors are inherently compatible with fibre optic microwave systems that can integrate with wireless antennas, and can provide connectivity with in-built signal conditioning and electromagnetic interference immunity. Recent methods in wideband and adaptive signal processing, which address the challenge of realising programmable microwave photonic phase shifters and true-time delay elements for phased array beamforming; ultra-wideband Hilbert transformers; single passband, widely tunable, and switchable microwave photonic filters; and ultrawideband microwave photonic mixers, are described. In addition, a new microwave photonic mixer structure is presented, which is based on using the inherent frequency selectivity of the stimulated Brillouin scattering loss spectrum to suppress the carrier of a dual-phase modulated optical signal. Results for the new microwave photonic mixer demonstrate an extremely wide bandwidth operation of 0.2 to 20 GHz and a large conversion efficiency improvement compared to the conventional microwave photonic mixer.