This paper reviews the demonstrations of photonic Hilbert transformers (PHTs), describing their progress and recent developments. The physical operating principles of PHTs including fractional Hilbert transformers are discussed, together with device applications in all-optical signal processing. Versatile approaches to realize PHTs are discussed, e.g., discrete free space optics, fiber-based schemes and integrated planar geometry. The numerical designs and experimental performances of these PHTs are analyzed in terms of spectral quality, operating bandwidth, system integration, and mechanical and thermal stability. Recent developments of the monolithically integrated photonic Hilbert transform (HT) devices include directional couplers and planar Bragg gratings which allow all-optical single-sideband (SSB) suppression and sideband switching.