The need for a new generation of robots able to safely locomote and manipulate beside or cooperatively with humans or in un-constructed environments has recently emerged as a priority of the robotics community. ln the past two decades, this challenge has been tackled by the growing field of soft robotics mostly focusing on developing completely deformable soft robotic bodies that can freely deform along any direction and comply with any unexpected or excessive external force. However, intrinsic softness can be a limitation in situations that require exerting substantial forces, to withstand bodyweights or to modulate forces applied to the environment. Inspiration for a novel mechanically hybrid approach in robotics can come from a novel model for biological structures based on the concept of tensegrity. This model rejects the idea of the vertebrate musculoskeletal system as beams, columns, and levers, and acknowledges the bones as stiff under compression elements held and stabilized in their positions by the pull of tensile components such as muscles, tendons, ligaments, and fascia. Therefore, rather than a frame supporting an amorphous soft tissue mass, the whole vertebrate body is divided into several modules stabilized by these tension components and its stiffness can vary depending on the pre-stress (i.e. tone) and stiffness of its muscles. This structural organization allows biological organisms to be more mechanically adaptive, robust, and lightweight at the same time. They can exhibit high physical compliance, apply efficient forces to the environment when needed, and, at the same time, withstand their bodyweight at different scales. In the same way, modular tensegrity structures used in robotic applications can provide a lightweight and robust framework where integrating stiff under compression struts for bodyweight support; variable stiffness cables for applying directionally larger forces when required, and soft cables to absorb energy and allow compliance. ln this thesis, tensegrity modular structures are investigated as a new approach to develop soft robots with variable stiffness capabilities. This investigation led to a set of design strategies to manufacture and actuate tensegrity modules with programmable stiffness and deformation; to mechanically and electrically connect tensegrity modules to develop untethered applications, and to implement variable-stiffness capabilities in the modules for active and passive stiffness change. Finally, preliminary results and insights are presented on using heuristic algorithms as a design method to find task-optimal morphologies, stiffness, and control of modular tensegrity robots. In this thesis modular tensegrity robots have been developed for performing locomotion and grasping tasks, however, the design methods presented can provide a guideline for designing robots for diverse applications where being lightweight, robust, and having variable-stiffness capabilities are of primary importance. We believe that the work presented and discussed in this thesis answered a gap in the current literature and most importantly, will pave the way to future research on tensegrity robots for a great variety of tasks and robotic applications.