PointCNN is a novel framework for feature learning from point clouds, addressing the challenges posed by the irregular and unordered nature of point clouds. The key innovation is the $\mathcal{X}$-transformation, which permutes and weights input features associated with points, followed by a typical convolution operation. This transformation is learned using a multilayer perceptron (MLP) to promote both weighting and permutation, ensuring that the output features are invariant to point ordering while leveraging spatially-local correlations. The proposed method, called PointCNN, generalizes traditional CNNs to handle point cloud data and achieves competitive or superior performance on multiple benchmark datasets and tasks compared to state-of-the-art methods. Experiments demonstrate that PointCNN effectively captures shape information and outperforms other point cloud processing methods in various applications, including classification, segmentation, and sketch recognition.PointCNN is a novel framework for feature learning from point clouds, addressing the challenges posed by the irregular and unordered nature of point clouds. The key innovation is the $\mathcal{X}$-transformation, which permutes and weights input features associated with points, followed by a typical convolution operation. This transformation is learned using a multilayer perceptron (MLP) to promote both weighting and permutation, ensuring that the output features are invariant to point ordering while leveraging spatially-local correlations. The proposed method, called PointCNN, generalizes traditional CNNs to handle point cloud data and achieves competitive or superior performance on multiple benchmark datasets and tasks compared to state-of-the-art methods. Experiments demonstrate that PointCNN effectively captures shape information and outperforms other point cloud processing methods in various applications, including classification, segmentation, and sketch recognition.