Holographic techniques are fundamental to applications such as volumetric displays1, high-density data storage and optical tweezers that require spatial control of intricate optical2 or acoustic fields3,4 within a three-dimensional volume. The basis of holography is spatial storage of the phase and/or amplitude profile of the desired wavefront5,6 in a manner that allows that wavefront to be reconstructed by interference when the hologram is illuminated with a suitable coherent source. Modern computer-generated holography7 skips the process of recording a hologram from a physical scene, and instead calculates the required phase profile before rendering it for reconstruction. In ultrasound applications, the phase profile is typically generated by discrete and independently driven ultrasound sources3,4,8,9,10,11,12; however, these can only be used in small numbers, which limits the complexity or degrees of freedom that can be attained in the wavefront. Here we introduce monolithic acoustic holograms, which can reconstruct diffraction-limited acoustic pressure fields and thus arbitrary ultrasound beams. We use rapid fabrication to craft the holograms and achieve reconstruction degrees of freedom two orders of magnitude higher than commercial phased array sources. The technique is inexpensive, appropriate for both transmission and reflection elements, and scales well to higher information content, larger aperture size and higher power. The complex three-dimensional pressure and phase distributions produced by these acoustic holograms allow us to demonstrate new approaches to controlled ultrasonic manipulation of solids in water, and of liquids and solids in air. We expect that acoustic holograms will enable new capabilities in beam-steering and the contactless transfer of power, improve medical imaging, and drive new applications of ultrasound.