Multiview three-dimensional (3D) displays can project the correct perspectives of a 3D image in many spatial directions simultaneously. They provide a 3D stereoscopic experience to many viewers at the same time with full motion parallax and do not require special glasses or eye tracking. None of the leading multiview 3D solutions is particularly well suited to mobile devices (watches, mobile phones or tablets), which require the combination of a thin, portable form factor, a high spatial resolution and a wide full-parallax view zone (for short viewing distance from potentially steep angles). Here we introduce a multi-directional diffractive backlight technology that permits the rendering of high-resolution, full-parallax 3D images in a very wide view zone (up to 180 degrees in principle) at an observation distance of up to a metre. The key to our design is a guided-wave illumination technique based on light-emitting diodes that produces wide-angle multiview images in colour from a thin planar transparent lightguide. Pixels associated with different views or colours are spatially multiplexed and can be independently addressed and modulated at video rate using an external shutter plane. To illustrate the capabilities of this technology, we use simple ink masks or a high-resolution commercial liquid-crystal display unit to demonstrate passive and active (30 frames per second) modulation of a 64-view backlight, producing 3D images with a spatial resolution of 88 pixels per inch and full-motion parallax in an unprecedented view zone of 90 degrees. We also present several transparent hand-held prototypes showing animated sequences of up to six different 200-view images at a resolution of 127 pixels per inch.