Breast cancer is the most common invasive cancer in women, affecting more than 10% of women worldwide. Microscopic analysis of a biopsy remains one of the most important methods to diagnose the type of breast cancer. This requires specialized analysis by pathologists, in a task that i) is highly time- and cost-consuming and ii) often leads to nonconsensual results. The relevance and potential of automatic classification algorithms using hematoxylin-eosin stained histopathological images has already been demonstrated, but the reported results are still sub-optimal for clinical use. With the goal of advancing the state-of-the-art in automatic classification, the Grand Challenge on BreAst Cancer Histology images (BACH) was organized in conjunction with the 15th International Conference on Image Analysis and Recognition (ICIAR 2018). A large annotated dataset, composed of both microscopy and whole-slide images, was specifically compiled and made publicly available for the BACH challenge. Following a positive response from the scientific community, a total of 64 submissions, out of 677 registrations, effectively entered the competition. From the submitted algorithms it was possible to push forward the state-of-the-art in terms of accuracy (87%) in automatic classification of breast cancer with histopathological images. Convolutional neuronal networks were the most successful methodology in the BACH challenge. Detailed analysis of the collective results allowed the identification of remaining challenges in the field and recommendations for future developments. The BACH dataset remains publically available as to promote further improvements to the field of automatic classification in digital pathology.