The popularity of mobile robots has been steadily growing, with these robots being increasingly utilized to execute tasks previously completed by human workers. For bipedal robots to see this same success, robust autonomous navigation systems need to be developed that can execute in real-time and respond to dynamic environments. These systems can be divided into three stages: perception, planning, and control. A holistic navigation framework for bipedal robots must successfully integrate all three components of the autonomous navigation problem to enable robust real-world navigation. In this paper, we present a real-time navigation framework for bipedal robots in dynamic environments. The proposed system addresses all components of the navigation problem: We introduce a depth-based perception system for obstacle detection, mapping, and localization. A two-stage planner is developed to generate collision-free trajectories robust to unknown and dynamic environments. And execute trajectories on the Digit bipedal robot's walking gait controller. The navigation framework is validated through a series of simulation and hardware experiments that contain unknown environments and dynamic obstacles.