Abstract: Photoacoustic Tomography (PAT) is an emerging biomedical "Imaging from Coupled Physics"-technique, in which contrast induced by one type of wave is read by another kind: A pulse of infra-red laser light is sent into biological tissue where it spreads until it is absorbed, whereupon it creates a local increase in pressure which propagates to the surface as a broadband, ultrasonic pulse. The initial pressure increase can be reconstructed from surface recordings with a high spatial resolution. In a second step, this internal information can be utilized to reconstruct the optical absorption coefficient, which provides a high contrast for various (pre-)clinical tasks. The talk will start with a general introduction to PAT before focusing on the various mathematical and computational challenges it involves. Finally, we discuss the additional difficulties of imaging fast dynamic processes with PAT and outline how we try to overcome them by combing recent advances in spatio-temporal sub-sampling schemes, compressed sensing and inverse problems with the development of tailored data acquisition systems.