Random planar maps and two-dimensional gravity

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The relationship between random planar geometries, two-dimensional quantum gravity and string theories has been studied by theoretical physicists and mathematicians since 35 years. I shall present recent works on random Strebel graphs and random Delaunay triangulations, and discuss their relations with the geometry of moduli spaces of surfaces, topological gravity, conformal point processes, and possible discretisations of conformal theories. Based on joint works with B. Eynard, S. Charbonnier and J. Scott