In various applications of airborne laser scanning (ALS), the classification of the point cloud is a basic and key step. It requires assigning category labels to each point, such as ground, building or vegetation. Convolutional neural networks have achieved great success in image classification and semantic segmentation, but they cannot be directly applied to point cloud classification because of the disordered and unstructured characteristics of point clouds. In this paper, we design a novel convolution operator to extract local features directly from unstructured points. Based on this convolution operator, we define the convolution layer, construct a convolution neural network to learn multi-level features from the point cloud, and obtain the category label of each point in an end-to-end manner. The proposed method is evaluated on two ALS datasets: the International Society for Photogrammetry and Remote Sensing (ISPRS) Vaihingen 3D Labeling benchmark and the 2019 IEEE Geoscience and Remote Sensing Society (GRSS) Data Fusion Contest (DFC) 3D dataset. The results show that our method achieves state-of-the-art performance for ALS point cloud classification, especially for the larger dataset DFC: we get an overall accuracy of 97.74% and a mean intersection over union (mIoU) of 0.9202, ranking in first place on the contest website.