Complete coverage navigation (CCN) requires a special type of robot path planning, where the robots should pass every part of the workspace. CCN is an essential issue for cleaning robots and many other robotic applications. When robots work in unknown environments, map building is required for the robots to effectively cover the complete workspace. Real-time concurrent map building and complete coverage robot navigation are desirable for efficient performance in many applications. In this paper, a novel neural-dynamics-based approach is proposed for real-time map building and CCN of autoxnomous mobile robots in a completely unknown environment. The proposed model is compared with a triangular-cell-map-based complete coverage path planning method (Oh , 2004) that combines distance transform path planning, wall-following algorithm, and template-based technique. The proposed method does not need any templates, even in unknown environments. A local map composed of square or rectangular cells is created through the neural dynamics during the CCN with limited sensory information. From the measured sensory information, a map of the robot's immediate limited surroundings is dynamically built for the robot navigation. In addition, square and rectangular cell map representations are proposed for real-time map building and CCN. Comparison studies of the proposed approach with the triangular-cell-map-based complete coverage path planning approach show that the proposed method is capable of planning more reasonable and shorter collision-free complete coverage paths in unknown environments.