This volume arises from a meeting organized at IHP in January 2007 in honor of the 80th birthday of Murray Gerstenhaber and the 70th birthday of Jim Stasheff. The meeting was centered on algebraic deformations and higher homotopies, seminal concepts pioneered by Gerstenhaber and Stasheff in the early 1960s that now are ubiquitous in fundamental areas of mathematics (algebra, algebraic topology, differential geometry, algebraic geometry, mathematical physics) and of theoretical physics (quantum field theory, string theory).