Goldstone's theorem (1962): When a continuous symmetry is spontaneously broken, there must exist a gapless (massless) mode — the Goldstone boson. In a crystal, translational symmetry is broken → acoustic phonons with ω→0 as k→0 (linear dispersion). In the XY model (spins free to rotate in a plane), U(1) symmetry is broken below T_KT (Kosterlitz-Thouless temperature) → spin-wave Goldstone modes with ω∝k². Adding explicit symmetry breaking (anisotropy, magnetic field) gaps out the mode: ω² = v²k² + m² (Higgs mechanism analogy). The Higgs boson is the massive partner of Goldstone modes when local gauge symmetry is considered. Pseudo-Goldstone bosons (pions in QCD) arise from approximate symmetry breaking.