Breast cancer is among the most common and fatal diseases for women, and no permanent treatment has been discovered. Thus, early detection is a crucial step to control and cure breast cancer that can save the lives of millions of women. For example, in 2020, more than 65% of breast cancer patients were diagnosed in early-stage of cancer, from whom all survived cancer. Although early detection is the most effective approach for cancer treatment, breast cancer screening conducted by radiologists is very expensive and time-consuming. More importantly, conventional methods of analyzing breast cancer images suffer from high false detection rates. Different breast cancer imaging modalities are used to extract and analyze the key features affecting diagnosis and treatment of breast cancer. These imaging modalities can be divided into subgroups such as mammograms, ultrasound, magnetic resonance imaging, histopathological images, or any combination of them. Radiologists or pathologists analyze images produced by these methods manually that leads to increase the risk of wrong decisions for cancer detection. Thus, utilization of new automatic methods to analyze all kinds of breast screening images to assist radiologists to interpret images is required. Recently, artificial intelligence (AI) has been widely utilized to automatically improve the early detection and treatment of different types of cancer, specifically breast cancer, thereby enhancing the survival chance of patients. Advances in AI algorithms, such as deep learning, and availability of datasets obtained from various imaging modalities have opened an opportunity to surpass limitations of current breast cancer analysis methods. In this article, we first review breast cancer imaging modalities, and their strengths and limitations. Then, we explore and summarize the most recent studies that employed AI in breast cancer detection using various breast imaging modalities. In addition, we report available datasets on the breast cancer imaging modalities which are important in developing AI-based algorithms and training deep learning models. In conclusion, this review paper tries to provide a comprehensive resource to help researchers working in breast cancer imaging analysis.