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In this collection of essays one finds contributions from Davide Buoso, Giuseppe Buttazzo, Michel Delfour, Gianpaolo Leonardi, Dario Mazzoleni, Al-hassem Nayam, Alberto Paganini, Volker Schulz, Jan Sokolowski, Kevin Sturm, Michiel Van den Berg, Bozhidar Velichkov and Alessandro Zilio. New trends in Shape Optimization was the title of a Workshop that we have organized in Erlangen in September 2013. We have had a very interesting week with many contributions by both senior mathematicians and young students; in particular, many open problems have been described, and a fruitful spontaneous...