In this paper, we introduce SalsaNext for the uncertainty-aware semantic segmentation of a full 3D LiDAR point cloud in real-time. SalsaNext is the next version of SalsaNet which has an encoder-decoder architecture where the encoder unit has a set of ResNet blocks and the decoder part combines upsampled features from the residual blocks. In contrast to SalsaNet, we introduce a new context module, replace the ResNet encoder blocks with a new residual dilated convolution stack with gradually increasing receptive fields and add the pixel-shuffle layer in the decoder. Additionally, we switch from stride convolution to average pooling and also apply central dropout treatment. To directly optimize the Jaccard index, we further combine the weighted cross entropy loss with Lovasz-Softmax loss. We finally inject a Bayesian treatment to compute the epistemic and aleatoric uncertainties for each point in the cloud. We provide a thorough quantitative evaluation on the Semantic-KITTI dataset, which demonstrates that the proposed SalsaNext outperforms other state-of-the-art semantic segmentation networks and achieves 3.6% more accuracy over the previous state-of-the-art method. We also release our source code at https://anonymous.4open.science/r/8f466bde-b2c0-4e49-9425-ecf7f0eb9c8f.