Traditional methods for computing quantities such as scattering amplitudes and form factors in quantum field theory become intractable at high perturbative orders. However, a great deal is now known about the mathematical properties of these quantities, especially in supersymmetric gauge theories. In this talk, I will describe how this knowledge can be leveraged to 'bootstrap' amplitudes and form factors directly, by constructing an ansatz with the appropriate mathematical structure and requiring it to have certain expected behavior in special kinematic limits. I will focus on the example of three-point form factors in maximally supersymmetric gauge theory, which have recently been bootstrapped through eight loops. I will then describe a remarkable new duality between this form factor and certain six-particle amplitudes in the same theory, which holds order by order in perturbation theory.