The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal processing. However, in many modern applications, the signal bandwidths have increased tremendously, while the acquisition capabilities have not scaled sufficiently fast. Consequently, conversion to digital has become a serious bottleneck. Furthermore, the resulting high rate digital data requires storage, communication and processing at very high rates which is computationally expensive and requires large amounts of power. In the context of medical imaging sampling at high rates often translates to high radiation dosages, increased scanning times, bulky medical devices, and limited resolution. In this talk, we present a framework for sampling and processing a wide class of wideband analog signals at rates far below Nyquist by exploiting signal structure and the processing task and show several demos of real-time sub-Nyquist prototypes. We then consider applications of these ideas to a variety of problems in medical and optical imaging including fast and quantitative MRI, wireless ultrasound, fast Doppler imaging, and correlation based super-resolution in microscopy and ultrasound which combines high spatial resolution with short integration time. We end by discussing several modern methods for structure-based phase retrieval which has applications in several areas of optical imaging.