Geometry emerging from spectra

We give a gentle introduction to the spectral approach to geometry, where we replace spaces by commutative algebras, and capture the metric by combining it with the vibration spectrum of a suitable operator on the geometric space. We will give many examples and also show how to reconstruct geometry from this spectral data. This will allow us to generalize to noncommutative spaces, which we illustrate by some of the key examples. We will conclude by establishing some convergence results on the emerging geometric spaces when an increasing part of the spectrum is available.