A Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined by the boundary distance function, that is the restriction of the distance function to the boundary. Loosely speaking, this means that the Riemannian metric can be recovered from measuring distances between boundary points only. The goal is to show that certain classes of metrics are boundary rigid (and, ideally, to suggest a procedure for recovering the metric). This problem has been extensively studied from PDE viewpoint: the distance between boundary points can be interpreted as a "travel time" for a solution of the wave equation. Hence this becomes a classic Inverse Problem when we have some information about solutions of a certain PDE and want to recover its coefficients. For instance such problems naturally arise in geophysics (when we want to find out what is inside the Earth by sending sound waves), medical imaging etc. In a joint project with S. Ivanov we suggest an alternative geometric approach to this problem. This approach grew from our long-term project of studying general surface area functionals and minimal surfaces in normed spaces. We have been working on it for more than ten years. In our earlier work we used this approach to show boundary rigidity for metrics close to flat ones (in all dimensions), thus giving the first open class of boundary rigid metrics beyond two dimensions. We were now able to extend this result to include metrics close to a hyperbolic one. This result is the main topic of the talk. The proof is based on a very transparent scheme, which however has a few blocks where certain formulas have to be carefully chosen, and this turns out to be a highly non-trivial task. The purpose of the talk is to present this scheme and to explain what is needed from those blocks to make them work. The talk assumes no background in inverse problems and is supposed to be "understandable" (in other words, we will not get into technical details of the proofs). We will also discuss several striking problems which seem to be widely open.