One of the major challenges in robotics and engineering is to develop efficient technological solutions that are able to cope with complex environments and unpredictable constraints. Taking inspiration from natural organisms is a well-known approach to tackling these issues. Climbing plants are an important, yet innovative, source of inspiration due to their ability to adapt to diverse habitats, and can be used as a model for developing robots and smart devices for exploration and monitoring, as well as for search and rescue operations. This review reports the main methodologies and approaches used by scientists to investigate and extract the features of climbing plants that are relevant to the artificial world in terms of adaptation, movement, and behaviour, and it summarizes the current available climbing plant-inspired engineering solutions.