The introduction of engineered resonance phenomena on surfaces has opened a new frontier in surface science and technology. Pillared phononic crystals, metamaterials, and metasurfaces are an emerging class of artificial structured media, featuring surfaces, that consist of pillars–or branching substructures–standing on a plate or a substrate. A pillared phononic crystal exhibits Bragg band gaps while a pillared metamaterial may feature both Bragg band gaps and local-resonance hybridization band gaps. These two band-gap phenomena, along with other unique wave dispersion characteristics, have been exploited for a variety of applications spanning a range of length scales and covering multiple disciplines in applied physics and engineering, particularly in elastodynamics and acoustics. The intrinsic placement of pillars on a semi-infinite surface–yielding a metasurface–has similarly provided new avenues for the control and manipulation of wave propagation. Classical waves are admitted in pillared media, including Lamb waves in plates and Rayleigh and Love waves along the surface of substrates, ranging in frequencies from Hz to several GHz. With the presence of the pillars, these waves couple with surface resonances richly creating new phenomena and properties in the subwavelength regime and in some applications at higher frequencies as well. At the nanoscale, it was shown that atomic-scale resonances–stemming from nanopillars–alter the fundamental nature of conductive thermal transport by reducing the group velocities and generating mode localizations across the entire spectrum of the constituent material well into the THz regime. In this article, we first overview the history and development of pillared materials, then provide a detailed synopsis of a selection of key research topics that involve the utilization of pillars or similar branching substructures in different contexts. Finally, we conclude by providing a short summary and some perspectives on the state of the field and its promise for further future development.