The multi-billion dollar, worldwide medical ultrasound (US) market continues to grow annually. Its non-ionizing nature, real-time capabilities and relatively low cost, compared to other imaging modalities, have led to significant applications in many different fields, including cardiology, angiology, obstetrics and emergency medicine. Facilitated by ongoing innovations, US continues to change rules and norms regarding patient screening, diagnosis and surgery. This huge and promising market is constantly driven by new imaging and processing techniques. From 3D images to sophisticated software, hardware and portability improvements, it is clear that the status of US as one of the leading medical imaging technologies is ensured for many years ahead. However, as imaging systems evolve, new engineering challenges emerge. Acquisition, transmission and processing of huge amounts of data are common for all ultrasound-based imaging modalities. Moreover, achieving higher resolution is constantly on demand, as improved diagnosis could be achieved by better visualization of organs and blood vessels deep within tissues. In this article, our goal is to motivate further interest and research in emerging processing techniques, as well as their applications in medical ultrasound, enabled by recent advancements in signal processing algorithms and deep learning. We address some of the primary challenges and potential remedies from a signal processing perspective, by exploiting the inherent structure of the received US signal.