When an integrable quantum field theory in one space dimension has a Lie group symmetry, the Lie algebra is typically embedded in a larger algebra called a Yangian. When one adds a boundary which preserves integrability, this is extended to a (generalized) twisted Yangian. We explain the role of these algebras in physics, and in particular recent work by MacKay and Regelskis which uncovers their governing role in the scattering of worldsheet excitations off D-branes in the AdS/CFT correspondence.