The papers in this volume are based on talks given at the 2001 Manchester Meeting of the London Mathematical Society, which was followed by an international workshop on 'Quantization, Deformations, and New Homological and Categorical Methods in Mathematical Physics'. The focus is on the topics suggested by the title: Quantization in its various aspects, Poisson brackets and generalizations, and structures 'beyond', including symplectic supermanifolds, operads, Lie groupoids and Lie (bi) algebroids and algebras with $n$-ary operations. This book offers accounts of new results as well as accessible expositions useful to a broad reading audience of researchers in differential geometry, algebraic topology and mathematical physics.