Explore the fundamentals of the analytic S-matrix bootstrap in this comprehensive lecture by Alexander Zhiboedov from CERN's Theory Division. Delve into the concept of solving theories through internal consistency, drawing parallels with the recent success of the conformal bootstrap. Review classic results and discover recent developments in both analytic and numerical approaches. Topics covered include the Mandelstam plane, scattering as a boundary value of analytic functions, maximal analyticity, Mandelstam representation, and the AKS theorem. Engage with exercises, explore key conditions and assumptions, and participate in a Q&A session to deepen your understanding of this complex subject in theoretical physics.