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3D object classification has attracted appealing attentions in academic researches and industrial applications. However, most existing methods need to access the training data of past 3D object classes when facing the common real-world scenario: new classes of 3D objects arrive in a sequence. Moreover, the performance of advanced approaches degrades dramatically for past learned classes (i.e., catastrophic forgetting), due to the irregular and redundant geometric structures of 3D point cloud data. To address these challenges, we propose a new Incremental 3D Object Learning (i.e., I3DOL) model, which is the first exploration to learn new classes of 3D object continually. Specifically, an adaptive-geometric centroid module is designed to construct discriminative local geometric structures, which can better characterize the irregular point cloud representation for 3D object. Afterwards, to prevent the catastrophic forgetting brought by redundant geometric information, a geometric-aware attention mechanism is developed to quantify the contributions of local geometric structures, and explore unique 3D geometric characteristics with high contributions for classes incremental learning. Meanwhile, a score fairness compensation strategy is proposed to further alleviate the catastrophic forgetting caused by unbalanced data between past and new classes of 3D object, by compensating biased prediction for new classes in the validation phase. Experiments on 3D representative datasets validate the superiority of our I3DOL framework.