We review the biomechanics, functional morphology, and physiology of motile traps. The movements of snap traps in Aldrovanda and Dionaea, motile adhesive traps in Drosera and Pinguicula, and suction traps in Utricularia are driven by active water displacement processes leading to reversible turgor changes of motor cells, irreversible growth, or mechanical pre-stressing of tissues. In some cases, the motion is amplified by the release of elastic energy stored in these tissues. The only known case of a passive motile trapping movement is the ‘springboard’ trapping mechanism of Nepenthes gracilis, in which a rapid vibration of the pitcher lid is actuated by the impact force of raindrops. Open research questions are summarized and future studies are suggested.