Breast cancer is the second-leading cause of mortality among women around the world. Ultrasound (US) is one of the noninvasive imaging modalities used to diagnose breast lesions and monitor the prognosis of cancer patients. It has the highest sensitivity for diagnosing breast masses, but it shows increased false negativity due to its high operator dependency. Underserved areas do not have sufficient US expertise to diagnose breast lesions, resulting in delayed management of breast lesions. Deep learning neural networks may have the potential to facilitate early decision-making by physicians by rapidly yet accurately diagnosing and monitoring their prognosis. This article reviews the recent research trends on neural networks for breast mass ultrasound, including and beyond diagnosis. We discussed original research recently conducted to analyze which modes of ultrasound and which models have been used for which purposes, and where they show the best performance. Our analysis reveals that lesion classification showed the highest performance compared to those used for other purposes. We also found that fewer studies were performed for prognosis than diagnosis. We also discussed the limitations and future directions of ongoing research on neural networks for breast ultrasound.