Keywords: Gromov-Hausdorff distance, Gromov-Wasserstein distance, shape analysis, metric geometry Abstract: The Gromov-Hausdorff distance provides a powerful tool for formalizing the problem of shape matching. Two problems with it are that (1) in practice it leads to combinatorial optimization problems which are NP hard and (2) despite its theoretical attractiveness and naturality, it has been difficult to use for studying and establishing links to the many other shape matching methods available in the literature. The Gromov-Wasserstein distance, a variant of the Gromov-Hausdorff distance that is based on mass transportation ideas, directly leads to continuous optimization problems with quadratic objective functions and linear constraints. Furthermore, it has been proved that the methods based on shape distributions, shape context/integral invariants, and eccentricity can all be related to this Gromov-Wasserstein distance via explicit lower bounds. In this talk we review the construction of the GW distance, its properties and lower bounds. In particular, we describe recent work done on (1) relating it to persistent topology based shape signatures, and (2) on defining a certain spectral notion of the GW distance. This spectral notion of the GW distance permits proving that several invariants constructed from the spectrum of the Laplace-Beltrami operator on manifolds are stable in a quantitative way.