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). BACH aimed at the classification and localization of clinically relevant histopathological classes in microscopy and whole-slide images from a large annotated dataset, specifically compiled and made publicly available for the challenge. Following a positive response from the scientific community, a total of 64 submissions, out of 677 registrations, effectively entered the competition. The submitted algorithms improved the state-of-the-art in automatic classification of breast cancer with microscopy images to an accuracy of 87%. 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 publicly available as to promote further improvements to the field of automatic classification in digital pathology.