Proof complexity Proof complexity is a rich subject drawing on methods from logic combinatorics algebra and computer science. This self contained book presents the basic concepts classical results current state of the art and possible future directions in the field. It stresses a view of proof complexity as a whole entity rather than a collection of various topics held together loosely by a few notions and it favors more generalizable statements. Lower bounds for lengths of proofs often regarded as the key issue in proof complexity are of course covered in detail. However upper bounds are not neglected: this book also explores the relations between bounded arithmetic theories and proof systems and how they can be used to prove upper bounds on lengths of proofs and simulations among proof systems. It goes on to discuss topics that transcend specific proof systems allowing for deeper understanding of the fundamental problems of the subject Concepts and problems Frege systems Sequent calculus Quantifed propositional calculus Resolution Algebraic and geometric proof systems Further proof systems Basic example of the correspondence Two worlds of bounded arithmetic Up to EF via the <...> translation Examples of upper bounds and p simulations Beyond EF via the || ... || translation R and R like proof systems LKD+1 2 and combinatorial restrictions Fd and logical restrictions Algebraic and geometric proof systems Feasible interpolation: a framework Feasible interpolation: applications Hard tautologies Model theory and lower bounds Optimality The nature of proof complexity