Using state-of-the-art deep learning models for cancer diagnosis presents several challenges related to the nature and availability of labeled histology images. In particular, cancer grading and localization in these images normally relies on both image- and pixel-level labels, the latter requiring a costly annotation process. In this survey, deep weakly-supervised learning (WSL) models are investigated to identify and locate diseases in histology images, without the need for pixel-level annotations. Given training data with global image-level labels, these models allow to simultaneously classify histology images and yield pixel-wise localization scores, thereby identifying the corresponding regions of interest (ROI). Since relevant WSL models have mainly been investigated within the computer vision community, and validated on natural scene images, we assess the extent to which they apply to histology images which have challenging properties, e.g. very large size, similarity between foreground/background, highly unstructured regions, stain heterogeneity, and noisy/ambiguous labels. The most relevant models for deep WSL are compared experimentally in terms of accuracy (classification and pixel-wise localization) on several public benchmark histology datasets for breast and colon cancer -- BACH ICIAR 2018, BreaKHis, CAMELYON16, and GlaS. Furthermore, for large-scale evaluation of WSL models on histology images, we propose a protocol to construct WSL datasets from Whole Slide Imaging. Results indicate that several deep learning models can provide a high level of classification accuracy, although accurate pixel-wise localization of cancer regions remains an issue for such images. Code is publicly available.