This keynote address will illustrate some aspects of current medical ultrasonic imaging and attempt to identify areas in which advanced signal processing may be able to contribute to the enhancement and extension of clinical ultrasound. Examples will be drawn primarily from echocardiography, in part because of challenges specifically associated with imaging the beating heart. The presentation will address the goals and needs of the (medical) user in order to avoid approaches which are technically elegant but clinically irrelevant. One feature peculiar to medical imaging, which limits the effective resolution to far less than the expected theoretical limit for the few hundred micrometer wavelength imaging systems in current use, is the subtle variation of elastic properties of soft tissue and the corresponding local variation in the speed of sound. Both random and systematic (for example, anisotropic) variations contribute to degradation in image quality, including the effect known as speckle. Opportunities for advanced signal processing may include not only approaches designed to enhance image resolution but also contributions to areas such as contrast agents and tissue characterization. The talk is designed to provide a broad overview which might serve as a common reference point for subsequent presentations.