The papers in this volume are based on talks given at the 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.