Biomimetics faces a continual challenge of how to bridge the gap between what Nature has so effectively evolved and the current tools and materials that engineers and scientists can exploit. Kirigami, from the Japanese ‘cut’ and ‘paper’, is a method of design where laminar materials are cut and then forced out-of-plane to yield 3D structures. Kirimimetic design provides a convenient and relatively closed design space within which to replicate some of the most interesting niche biological mechanisms. These include complex flexing organelles such as cilia in algae, energy storage and buckled structures in plants, and organic appendages that actuate out-of-plane such as the myoneme of the Vorticella protozoa. Where traditional kirigami employs passive materials which must be forced to transition to higher dimensions, we can exploit planar smart actuators and artificial muscles to create self-actuating kirigami structures. Here we review biomimetics with respect to the kirigami design and fabrication methods and examine how smart materials, including electroactive polymers and shape memory polymers, can be used to realise effective biomimetic components for robotic, deployable structures and engineering systems. One-way actuation, for example using shape memory polymers, can yield complete self-deploying structures. Bi-directional actuation, in contrast, can be exploited to mimic fundamental biological mechanisms such as thrust generation and fluid control. We present recent examples of kirigami robotic mechanisms and actuators and discuss planar fabrication methods, including rapid prototyping and 3D printing, and how current technologies, and their limitations, affect Kirigami robotics.