In recent years, convolutional neural networks for semantic segmentation of breast ultrasound (BUS) images have shown great success; however, two major challenges still exist. 1) Most current approaches inherently lack the ability to utilize tissue anatomy, resulting in misclassified image regions. 2) They struggle to produce accurate boundaries due to the repeated down-sampling operations. To address these issues, we propose a novel breast anatomy-aware network for capturing fine image details and a new smoothness term that encodes breast anatomy. It incorporates context information across multiple spatial scales to generate more accurate semantic boundaries. Extensive experiments are conducted to compare the proposed method and eight state-of-the-art approaches using a BUS dataset with 325 images. The results demonstrate the proposed method significantly improves the segmentation of the muscle, mammary, and tumor classes and produces more accurate fine details of tissue boundaries.