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Medical ultrasound imaging is an ongoing research field for digital signal processing. Following decades of developement in the analogue domain, the introduction of high power computation has led to increased activity and research in the fields of digital signal processing, and, most recently, in machine learning, for the sake of delivering higher quality imaging, while reducing the size of the data required to acquire, process, display and, recently, wirelessly transmit the data to remote devices. An overview of the basics of ultrasound acquisition and imaging is given, followed by a presentation of its limitations and challenges. An in-depth explanation about the development and algorithmics of the methods designed to address each of these challenges follows, with an introduction of the outstanding tasks and issues posed in each field, and their potential applications and benefits to modern healthcare.