Computed tomography, a major technique of medical and geophysical imaging and industrial non-destructive testing, has been a constant source of wonderful mathematical problems for the last several decades. It involves many deep mathematical techniques, most notably coming from integral geometry (Radon transform), partial differential equations, harmonic analysis, and numerical analysis. In the last several years, a new brand of medical imaging modalities has been emerging, a distinguishing feature of which is combining different types of signals (e.g., electromagnetic and ultrasound). The most developed among them, albeit still not in clinics, is the so called thermoacoustic (also known as photoacoustic or optoacoustic) tomography (TAT). The mathematical problems arising in TAT are complex and require a wide variety of analytic tools. The talk will survey the main problems of TAT and recent progress in this area. No prior knowledge of the subject will be assumed.