  When global continuous symmetries are spontaneously broken, there appear
gapless collective excitations called Nambu-Goldstone modes (NGMs) that govern
the low-energy property of the system. The application of this famous theorem
ranges from high-energy, particle physics to condensed matter and atomic
physics. When a symmetry breaking occurs in systems that lack the Lorentz
invariance to start with, as is usually the case in condensed matter systems,
the number of resulting NGMs can be fewer than that of broken symmetry
generators, and the dispersion of NGMs is not necessarily linear. In this
article, we review recently established formulas for NGMs associated with
broken internal symmetries that work equally for relativistic and
nonrelativistic systems. We also discuss complexities of NGMs originating from
space-time symmetry breaking. In the process we cover many illuminating
examples from various context. We also present a complementary point of view
from the Lieb-Schultz-Mattis theorem.
