Curvatures are ubiquitous in the animal kingdom — the spiral shell of the nautilus and the corkscrew horns of the blackbuck being iconic examples. Dynamic changes in curvature (i.e., curving) are most striking in the locomotion of some animal species — swimming in fishes and mollusks, looping in leeches, undulatory locomotion in snakes and lampreys, and also sperm motility through flagellum beating. When it comes to plants, which are sessile organisms with a rigid body, the terms ‘curvature’ and ‘curving’ evoke very different images — leaves of grass swaying in the breeze, a trunk dangerously bent by a powerful gust of wind, a branch sagging under the weight of its own fruits, as well as the frail arabesques of twining plants like the morning glory and the ivy, which were so influential in the Art Nouveau movement and many other artistic traditions. These various vegetal curves not only prompt creative inspiration in the mind of the beholder, they also initiate signaling cascades leading to developmental responses of the plant. Conversely, curvature can result from a biologically active process in response to an internal or external stimulus. Active curving or decurving are indeed important aspects of the plastic interplay of the developmental program of plants and their environment. Although easily accessible to observation, curvature and curving have only recently become the focus of active research in plant development. Lying at the nexus of biology, physics and mathematics, they require an interdisciplinary approach. The aim of this primer is to give readers an intuitive but accurate understanding of what curvature and curving are, as observed in the plant kingdom, then a more formal definition. We will discuss their role in plant development, both as a signal and as a response, and finally the practical issues and solutions involved in measuring plant curvatures.