GrowBot proposes a disruptively new paradigm of movement in robotics inspired by the moving-by-growing abilities of climbing plants. Plants are still a quite unexplored model in robotics and ICT technologies, as their sessile nature leads to think that they do not move. Instead, they move greatly, on a different time scale, purposively, effectively and efficiently. To move from one point to another, plants must grow and continuously adapt their body to the external environmental conditions. This continuous growth is particularly evident in climbing plants. By imitating them, the GrowBot objective is to develop low-mass and low-volume robots capable of anchoring themselves, negotiating voids, and more generally climbing, where current climbing robots based on wheels, legs, or rails would get stuck or fall. Specifically, the ability to grow will be translated by additive manufacturing processes inside the robot, which creates its body by depositing new materials with multi-functional properties, on the basis of the perceived external stimuli (without a pre-defined design). Energy efficiency will be intrinsic to such approach, but novel bio-hybrid energy harvesting solutions will be also implemented to generate energy by interfacing soft technologies with real plants. Perception and behavior will be based on the adaptive strategies that allow climbing plants to explore the environment, described mathematically after experimental observations. GrowBot would contribute to consolidate this ground-breaking and pioneering research area on plant-inspired robotics that, although still in its infancy, can represent a revolutionary approach in robotics, as it has already happened with plant-inspired solutions in material science. GrowBot is based on a strongly interdisciplinary character and can open the way for a new technological paradigm around the concept of growing robots, fostering a European innovation eco-system for several high-tech sectors.