Abstract:In the cancer diagnosis pipeline, digital pathology plays an instrumental role in the identification, staging, and grading of malignant areas on biopsy tissue specimens. High resolution histology images are subject to high variance in appearance, sourcing either from the acquisition devices or the H\&E staining process. Nuclei segmentation is an important task, as it detects the nuclei cells over background tissue and gives rise to the topology, size, and count of nuclei which are determinant factors for cancer detection. Yet, it is a fairly time consuming task for pathologists, with reportedly high subjectivity. Computer Aided Diagnosis (CAD) tools empowered by modern Artificial Intelligence (AI) models enable the automation of nuclei segmentation. This can reduce the subjectivity in analysis and reading time. This paper provides an extensive review, beginning from earlier works use traditional image processing techniques and reaching up to modern approaches following the Deep Learning (DL) paradigm. Our review also focuses on the weak supervision aspect of the problem, motivated by the fact that annotated data is scarce. At the end, the advantages of different models and types of supervision are thoroughly discussed. Furthermore, we try to extrapolate and envision how future research lines will potentially be, so as to minimize the need for labeled data while maintaining high performance. Future methods should emphasize efficient and explainable models with a transparent underlying process so that physicians can trust their output.