Quantifying forest biomass stocks and their dynamics is important for implementing effective climate change mitigation measures by aiding local forest management, studying processes driving af-, re-, and deforestation, and improving the accuracy of carbon accounting. Owing to the 3-dimensional nature of forest structure, remote sensing using airborne LiDAR can be used to perform these measurements of vegetation structure at large scale. Harnessing the full dimensionality of the data, we present deep learning systems predicting wood volume and above ground biomass (AGB) directly from the full LiDAR point cloud and compare results to state-of-the-art approaches operating on basic statistics of the point clouds. For this purpose, we devise different neural network architectures for point cloud regression and evaluate them on remote sensing data of areas for which AGB estimates have been obtained from field measurements in the Danish national forest inventory. Our adaptation of Minkowski convolutional neural networks for regression give the best results. The deep neural networks produce significantly more accurate wood volume, AGB, and carbon stock estimates compared to state-of-the-art approaches. In contrast to other methods, the proposed deep learning approach does not require a digital terrain model and is robust to artifacts along the boundaries of the evaluated areas, which we demonstrate for the case where trees protrude into the area from the outside. We expect this finding to have a strong impact on LiDAR-based analyses of biomass dynamics.