Positive and Unlabeled learning (PU learning) aims to build a binary classifier where only positive and unlabeled data are available for classifier training. However, existing PU learning methods all work on a batch learning mode, which cannot deal with the online learning scenarios with sequential data. Therefore, this paper proposes a novel positive and unlabeled learning algorithm in an online training mode, which trains a classifier solely on the positive and unlabeled data arriving in a sequential order. Specifically, we adopt an unbiased estimate for the loss induced by the arriving positive or unlabeled examples at each time. Then we show that for any coming new single datum, the model can be updated independently and incrementally by gradient based online learning method. Furthermore, we extend our method to tackle the cases when more than one example is received at each time. Theoretically, we show that the proposed online PU learning method achieves low regret even though it receives sequential positive and unlabeled data. Empirically, we conduct intensive experiments on both benchmark and real-world datasets, and the results clearly demonstrate the effectiveness of the proposed method.