A modular self-reconfigurable robot (MSRR) is a set of robotic building blocks that can be connected together in different ways. By rearranging these connections, the robot can adapt its shape to address a wider variety of tasks than a robot with a fixed morphology. However, traditional modular architectures fail to scale up to address large-scale and high-force applications. To meet these challenges, we introduce the first self-reconfigurable modular truss robot system. The Variable Topology Truss (VTT) is a synthesis of two robot paradigms: MSRRs and variable geometry truss robots (VGTs). The structural efficiency of the truss architecture alleviates scaling issues with modular robots, while the task flexibility afforded by reconfiguration expands the possible applications of VGT robots. We first demonstrate hardware prototypes of the components necessary to build a VTT, which include a novel reconfigurable spherical joint and an improved version of a high extension ratio linear actuator. Then, we characterize the reconfiguration capability afforded by the new mechanisms and apply graph-theoretic techniques to enumerate all possible reconfigurable truss topologies up to a certain size. With these techniques, we can identify potential sequences of actions that reconfigure between desired start and goal topologies. However, the entangled nature of the truss architecture presents new challenges for collision-free motion planning. The self-collision constraints divide up the configuration space into many disconnected regions. We develop a mathematical invariant inspired by knot theory---the link-augmented graph---which serves as a test to quickly prove when certain configurations lie in disconnected regions. This invariant can be combined with more traditional planning techniques to boost their performance on truss-like robots, which we demonstrate with a new variant of RRT-Connect. Finally, we combine the topological reconfiguration planning with the link-augmented graph and geometric planner to create a multi-modal planner for VTT. This planner is capable of finding a sequence of collision-free paths and reconfiguration actions that transform a VTT from a given start configuration to a goal configuration.