The purpose of this mini-course is to present a microlocal approach to

multi-wave imaging, including thermo- and photo-acoustic tomography. The mathematical model is an inverse source problem for the acoustic equation. We assume a variable sound speed. We will review first the theory of the wave equation and its microlocal parametrix. Then we will show how to get sharp uniqueness results for full and partial boundary observations using unique continuation. Next, we will study the stability problem with full and partial data.

In brain imaging, the speed is piecewise smooth only. This changes the

propagation of singularities and created new challenges. We will review the recent progress about this case as well.

Numerical simulations will be shown as well. The mini-course is based on joint papers with Gunther Uhlmann and the numerical results are obtained together with Uhlmann, Qian and Zhao.