The neuroscience of spatial memory is one of the most promising areas for developing biomimetic solutions to complex engineering challenges. Grid cells are neurons recorded in the medial entorhinal cortex that fire when rats are in an array of locations in the environment falling on the vertices of tightly packed equilateral triangles. Grid cells suggest an exciting new approach for enhancing robot simultaneous localization and mapping (SLAM) in changing environments and could provide a common map for situational awareness between human and robotic teammates. Current models of grid cells are well suited to robotics, as they utilize input from self-motion and sensory flow similar to inertial sensors and visual odometry in robots. Computational models, supported by in vivo neural activity data, demonstrate how grid cell representations could provide a substrate for goal-directed behavior using hierarchical forward planning that finds novel shortcut trajectories in changing environments.