The main topic will be inverse problems for linear and nonlinear wave equations. I will describe results in both stationary and non-stationary spacetimes. An example of inverse problems in stationary spacetimes is the imaging of internal structure of the earth from surface measurements of seismic waves arising from earthquakes or artificial explosions. Here, the materialistic properties of the internal layers of the earth are generally assumed to be independent of time. On the other hand, inverse problems for non-stationary spacetimes are inspired by the theory of general relativity as well as gravitational waves where waves follow paths that curve not only in space but also in time. We introduce a method of solving such inverse boundary value problems, and show that lower order coefficients can be recovered under certain curvature bounds. The talk is based on joint works with Spyros Alexakis and Lauri Oksanen.