Curriculum learning (CL) is referred to as a training strategy that makes easy samples learned first and then fits hard samples. It imitates the process of humans learning knowledge, and has become a potential manner of effectively training deep networks. In this study, we develop the adaptively point-weighting (APW) curriculum learning algorithm, which adaptively assigns the weight to every training sample not only based on its training error but also considering the current training state of the network. Specifically, in the early training phase, it increases the weights of easy samples to make the network rapidly capture the overall characteristics of the dataset; and in the later training phase, the weights of hard points rise to improve the fitting performance on the discrete local regions. Moreover, we also present the theoretical analysis on the properties of APW including training effectiveness, training feasibility, training stability, and generalization performance. The numerical experiments support the superiority of APW and demonstrate the validity of our theoretical findings.