Acoustic metamaterials consisting in massive materials perforated by periodic subwavelength holes or more sparse structures involving periodic arrangement of wires have been shown to be able to control the wave propagation with high flexibility. In comparison, phononic crystals have a priori higher dimensions because of their wavelength-scale period. However, if the metamaterial has subwavelength period, many of the observed phenomena are attributable to Fabry-Perot type resonances, resonances in the hole or resonances of the wires. Therefore, these structures have a limitation in their thickness, which has to be at wavelength scale to produce such resonances (and thickness refers to the size in the direction perpendicular to the plane containing the periodic cells). In order to reduce the size of the devices, structures with subwavelength thickness have been developed. They are known as metasurfaces and metafilms. Despite the vanishing thickness in comparison to the incident wavelength, the capability of these ultrathin devices to control the wave propagation has been evidenced. This is because they are based on a resonance which is not related to their thickness. It can be a thin elastic membrane within the unit cell, or resonances of labyrinthine or curled elements squeezed in the unit cell.