Knowledge of forest biomass stocks and their development is important for implementing effective climate change mitigation measures. Remote sensing using airborne LiDAR can be used to measure vegetation structure at large scale. We present deep learning systems for predicting wood volume, above-ground biomass (AGB), and subsequently above-ground carbon stocks directly from airborne LiDAR point clouds. Specifically, 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 a national forest inventory. Our adaptation of Minkowski convolutional neural networks for regression gave the best results. The deep neural networks produced significantly more accurate wood volume, AGB, and carbon estimates compared to state-of-the-art approaches operating on basic statistics of the point clouds. In contrast to other methods, no digital terrain model is required. We expect this finding to have a strong impact on LiDAR-based analyses of terrestrial ecosystem dynamics.