Human motion prediction is essential for tasks such as human motion analysis and human-robot interactions. Most existing approaches have been proposed to realize motion prediction. However, they ignore an important task, the evaluation of the quality of the predicted result. It is far more enough for current approaches in actual scenarios because people can't know how to interact with the machine without the evaluation of prediction, and unreliable predictions may mislead the machine to harm the human. Hence, we propose an uncertainty-aware framework for human motion prediction (UA-HMP). Concretely, we first design an uncertainty-aware predictor through Gaussian modeling to achieve the value and the uncertainty of predicted motion. Then, an uncertainty-guided learning scheme is proposed to quantitate the uncertainty and reduce the negative effect of the noisy samples during optimization for better performance. Our proposed framework is easily combined with current SOTA baselines to overcome their weakness in uncertainty modeling with slight parameters increment. Extensive experiments also show that they can achieve better performance in both short and long-term predictions in H3.6M, CMU-Mocap.