The counting task, which plays a fundamental rule in numerous applications (e.g., crowd counting, traffic statistics), aims to predict the number of objects with various densities. Existing object counting tasks are designed for a single object class. However, it is inevitable to encounter newly coming data with new classes in our real world. We name this scenario as \textit{evolving object counting}. In this paper, we build the first evolving object counting dataset and propose a unified object counting network as the first attempt to address this task. The proposed model consists of two key components: a class-agnostic mask module and a class-increment module. The class-agnostic mask module learns generic object occupation prior via predicting a class-agnostic binary mask (e.g., 1 denotes there exists an object at the considering position in an image and 0 otherwise). The class-increment module is used to handle new coming classes and provides discriminative class guidance for density map prediction. The combined outputs of class-agnostic mask module and image feature extractor are used to predict the final density map. When new classes come, we first add new neural nodes into the last regression and classification layers of this module. Then, instead of retraining the model from scratch, we utilize knowledge distilling to help the model remember what have already learned about previous object classes. We also employ a support sample bank to store a small number of typical training samples of each class, which are used to prevent the model from forgetting key information of old data. With this design, our model can efficiently and effectively adapt to new coming classes while keeping good performance on already seen data without large-scale retraining. Extensive experiments on the collected dataset demonstrate the favorable performance.