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This paper introduces a novel and general real-time adaptive motion planning (RAMP) approach suitable for planning trajectories of high-DOF or redundant robots, such as mobile manipulators, in dynamic environments with moving obstacles of unknown trajectories. The RAMP approach enables simultaneous path and trajectory planning and simultaneous planning and execution of motion in real time. It facilitates real-time optimization of trajectories under various optimization criteria, such as minimizing energy and time and maximizing manipulability. It also accommodates partially specified task goals of robots easily. The approach exploits redundancy in redundant robots (such as locomotion versus manipulation in a mobile manipulator) through loose coupling of robot configuration variables to best achieve obstacle avoidance and optimization objectives. The RAMP approach has been implemented and tested in simulation over a diverse set of task environments, including environments with multiple mobile manipulators. The results (and also the accompanying video) show that the RAMP planner, with its high efficiency and flexibility, not only handles a single mobile manipulator well in dynamic environments with various obstacles of unknown motions in addition to static obstacles, but can also readily and effectively plan motions for each mobile manipulator in an environment shared by multiple mobile manipulators and other moving obstacles.