Semantic feature learning on 3D point clouds is quite challenging because of their irregular and unordered data structure. In this paper, we propose a novel structure-aware convolution (SAC) to generalize deep learning on regular grids to irregular 3D point clouds. Similar to the template-matching process of convolution on 2D images, the key of our SAC is to match the point clouds’ neighborhoods with a series of 3D kernels, where each kernel can be regarded as a “geometric template” formed by a set of learnable 3D points. Thus, the interested geometric structures of the input point clouds can be activated by the corresponding kernels. To verify the effectiveness of the proposed SAC, we embedded it into three recently developed point cloud deep learning networks (PointNet, PointNet++, and KCNet) as a lightweight module, and evaluated its performance on both classification and segmentation tasks. Experimental results show that, benefiting from the geometric structure learning capability of our SAC, all these back-end networks achieved better classification and segmentation performance (e.g., +2.77% mean accuracy for classification and +4.99% mean intersection over union (IoU) for segmentation) with few additional parameters. Furthermore, results also demonstrate that the proposed SAC is helpful in improving the robustness of networks with the constraints of geometric structures.