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 Séminaires et Congrès 2013; 279 pp; softcover Number: 26 ISBN-13: 978-2-85629-363-8 List Price: US$64 Member Price: US$51.20 Order Code: SECO/26 An operad is a mathematical device used to encode universally a wide variety of algebraic structures. The name operad appeared first in the 1970s in algebraic topology to recognize $$n$$-fold loop spaces. Operads enjoyed a renaissance in the nineties, mainly under the impulse of quantum field theories. This universal notion is now used in many domains of mathematics such as differential geometry (deformation theory), algebraic geometry (moduli spaces of curves, Gromov-Witten invariants), noncommutative geometry (cyclic homology), algebraic combinatorics (Hopf algebras), theoretical physics (field theories, renormalization), computer science (rewriting systems) and universal algebra. The purpose of this volume is to present contributions about the notion of operads in these fields, where they play an important role. This volume is a result of a school and a conference, "Operads 2009", both of which took place at the CIRM (Luminy, France) in April 2009. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in algebraic theory. Table of Contents M. Batanin, C. Berger, and M. Markl -- Operads of natural operations I: Lattice paths, braces and Hochschild cochains F. Chapoton -- Categorification of the dendriform operad V. Dotsenko -- Freeness theorems for operads via Gröbner bases V. Dotsenko and M. V. Johansson -- Implementing Gröbner bases for operads B. Fresse -- Batanin's category of pruned trees is Koszul Y. Guiraud and P. Malbos -- Identities among relations for higher-dimensional rewriting systems Y. Lafont -- Diagram rewriting and operads Y. I. Manin -- Renormalization and computation I: Motivation and background T. Schedler -- Connes-Kreimer quantizations and PBW theorems for pre-Lie algebras D. P. Sinha -- The (non-equivariant) homology of the little disks operad $$A$$