  Using Bethe's hypothesis, C N Yang exactly solved the one-dimensional (1D)
delta-function interacting spin-1/2 Fermi gas with an arbitrary spin-imbalance
in 1967. At that time, using a different method, M Gaudin solved the problem of
interacting fermions in a spin-balanced case. Later, the 1D delta-function
interacting fermion problem was named as the Yang-Gaudin model. It has been in
general agreed that a key discovery of C N Yang's work was the cubic matrix
equation for the solvability conditions. % This equation was later
independently found by R J Baxter for commuting transfer matrices of 2D exactly
solvable vertex models. % The equation has since been referred to Yang-Baxter
equation, being the master equation to integrability. % The Yang-Baxter
equation has been used to solve a wide range of 1D many-body problems in
physics, such as 1D Hubbard model, $SU(N)$ Fermi gases, Kondo impurity problem
and strongly correlated electronic systems etc.
  % In this paper, we will briefly discuss recent developments of the
Yang-Gaudin model on several breakthroughs of many-body phenomena, ranging from
the universal thermodynamics to the Luttigner liquid, the spin charge
separation, the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like pairing state and
the quantum criticality.
  %
  These developments demonstrate that the Yang-Gaudin model has laid out a
profound legacy of the Yang-Baxter equation.
