Breast cancer is a prevalent health issue among women, with one in eight women succumbing to the disease. A significant number of women neglect the necessity for breast cancer detection, as the treatment poses risks associated with exposure to radioactive radiation. Non-invasive procedures, hazardous radiation exposure, and limited diagnostic specificity for breast tumors hinder breast cancer screening methods. Currently, several medical technologies, including mammography, magnetic resonance imaging, computed tomography, positron emission tomography, and histopathological imaging, are being used to diagnose breast cancer early. Nevertheless, proficient radiologists or pathologists are necessary to interpret these imaging modalities. The process is challenging, costly, and prone to inaccuracies. Advances in machine learning and computing technology have transformed many facets of the world in the last ten years. Deep learning models have emerged as powerful tools in detecting tumors and predicting breast cancer, leveraging radiographic and histopathological images to achieve remarkable outcomes. Nevertheless, rigorous external validation is essential to ensure these advanced artificial intelligence technologies can be confidently integrated into clinical decision-making processes. The main objective of this research is to provide a critical analysis of findings on breast cancerous tumor classification using machine and deep learning algorithms. For this, we examined five prominent datasets: Wisconsin, SEER, ultrasound pictures, mammograms, and BreakHis histopathology images. The study reviewed the literature from the past decade (2015–2024) across multiple sources, including Springer, ScienceDirect, IEEE, PubMed, MDPI, Nature, Web of Science, Hindawi, and ArXiv. A detailed discussion was incorporated to elucidate ongoing research challenges and opportunities for future research in this burgeoning field.