Deep learning (DL) is one of the most powerful data-driven machine-learning techniques in artificial intelligence (AI). It can automatically learn from raw data without manual feature selection. DL models have led to remarkable advances in data extraction and analysis for medical imaging. Magnetic resonance imaging (MRI) has proven useful in delineating the characteristics and extent of breast lesions and tumors. This review summarizes the current state-of-the-art applications of DL models in breast MRI. Many recent DL models were examined in this field, along with several advanced learning approaches and methods for data normalization and breast and lesion segmentation. For clinical applications, DL-based breast MRI models were proven useful in five aspects: diagnosis of breast cancer, classification of molecular types, classification of histopathological types, prediction of neoadjuvant chemotherapy response, and prediction of lymph node metastasis. For subsequent studies, further improvement in data acquisition and preprocessing is necessary, additional DL techniques in breast MRI should be investigated, and wider clinical applications need to be explored.