With the emergence of collaborative robots (cobots), human-robot collaboration in industrial manufacturing is coming into focus. For a cobot to act autonomously and as an assistant, it must understand human actions during assembly. To effectively train models for this task, a dataset containing suitable assembly actions in a realistic setting is cru-cial. For this purpose, we present the ATTACH dataset, which contains 51.6 hours of assembly with 95.2k annotated fine-grained actions monitored by three cameras, which represent potential viewpoints of a cobot. Since in an assembly context workers tend to perform different actions simultaneously with their two hands, we annotated the performed actions for each hand separately. Therefore, in the ATTACH dataset, more than 68% of annotations overlap with other annotations, which is many times more than in related datasets, typically featuring more simplistic assembly tasks. For better generalization with respect to the background of the working area, we did not only record color and depth images, but also used the Azure Kinect body tracking SDK for estimating 3D skeletons of the worker. To create a first baseline, we report the performance of state-of-the-art methods for action recognition as well as action detection on video and skeleton-sequence inputs. The dataset is available at https://www.tu-ilmenau.de/neurob/data-sets-code/attach-dataset.