Plants move in very different ways and for different reasons, but some active carnivorous plants perform extraordinary motion: Their snap-, catapult- and suction traps perform very fast and spectacular motions to catch their prey after receiving mechanical stimuli. Numerous investigations have led to deeper insights into the physiology and biomechanics of these trapping devices, but they are far from being fully understood. We review concisely how plant movements are classified and how they follow principles that bring together speed, actuation and architecture of the moving organ. In particular, we describe and discuss how carnivorous plants manage to execute fast motion. We address open questions and assess the prospects for future studies investigating potential universal mechanisms that could be the basis of key characteristic features in plant movement such as stimulus transduction, post-stimulatory mechanical answers, and organ formation.