The capability to sense and respond to external mechanical stimuli at various timescales is essential to many physiological aspects in plants, including selfprotection, intake of nutrients and reproduction. Remarkably, some plants have evolved the ability to react to mechanical stimuli within a few seconds despite a lack of muscles and nerves. The fast movements of plants in response to mechanical stimuli have long captured the curiosity of scientists and engineers, but the mechanisms behind these rapid thigmonastic movements are still not understood completely. In this article, we provide an overview of such thigmonastic movements in several representative plants, including Dionaea, Utricularia, Aldrovanda, Drosera and Mimosa. In addition, we review a series of studies that present biomimetic structures inspired by fast-moving plants. We hope that this article will shed light on the current status of research on the fast movements of plants and bioinspired structures and also promote interdisciplinary studies on both the fundamental mechanisms of plants' fast movements and biomimetic structures for engineering applications, such as artificial muscles, multi-stable structures and bioinspired robots.