With rapid advances in optical fabrication, digital processing power, and computational perception, a new generation of display technology is emerging: compressive displays exploring the co-design of optical elements and computational processing while taking particular characteristics of the human visual system into account. We will review advances in this field and give an outlook on next-generation compressive display and imaging technology. In contrast to conventional technology, compressive displays aim for a joint-design of optics, electronics, and computational processing that together exploit compressibility of the presented data. For instance, light fields show the same 3D scene from different perspectives - all these images are very similar and therefore compressible. By combining displays that use multilayer architectures or directional backlighting combined with optimal light field factorizations, limitations of existing devices, for instance resolution, depth of field, and field of view, can be overcome. In addition to light field display and projection, we will discuss a variety of technologies for compressive super-resolution and high dynamic range image display as well as compressive light field imaging and microscopy.