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This volume is an issue of the Rendiconti del Circolo Matematico di Palermo connected to the workshop “Recent advances in Linear series and Newton–Okounkov bodies” which took place at the Department of Mathematics of the University of Padova, Italy, in February, 9–14, 2015. We thank the Editors of the Rendiconti for having accepted to dedicate this issue to this topic. The workshop was organized by Ciro Ciliberto, Alex Küronya, Victor Lozovanu and Stefano Urbinati with the support of Foundation Compositio Mathematica, INDAM-GNSAGA, GRIFGA, Università degli Studi di Padova and Università degli Studi di Roma “Tor Vergata” and the National Research Project (PRIN) “Geometry of Algebraic Varieties”. The aim of the workshop was to bring together experts in the field of linear systems, Newton–Okounkov bodies and Nagata Conjecture to have a comparison of production in different research directions. There have been about 50 participants, both experts and young researchers in Algebraic Geometry, joining the workshop from all around the world. The structure of the workshop was very dynamic. There have been a number of introductory talks in the mornings, whose goal was to stimulate further discussions, combined with some more advanced talks dedicated to more recent advances. In the afternoons there has

Recent advances in linear series and Newton–Okounkov bodies