  Symmetry plays a fundamental role in physics. The quasi-degeneracy between
single-particle orbitals $(n, l, j = l + 1/2)$ and $(n-1, l + 2, j = l + 3/2)$
indicates a hidden symmetry in atomic nuclei, the so-called pseudospin symmetry
(PSS). Since the introduction of the concept of PSS in atomic nuclei, there
have been comprehensive efforts to understand its origin. Both splittings of
spin doublets and pseudospin doublets play critical roles in the evolution of
magic numbers in exotic nuclei discovered by modern spectroscopic studies with
radioactive ion beam facilities. Since the PSS was recognized as a relativistic
symmetry in 1990s, many special features, including the spin symmetry (SS) for
anti-nucleon, and many new concepts have been introduced. In the present
Review, we focus on the recent progress on the PSS and SS in various systems
and potentials, including extensions of the PSS study from stable to exotic
nuclei, from non-confining to confining potentials, from local to non-local
potentials, from central to tensor potentials, from bound to resonant states,
from nucleon to anti-nucleon spectra, from nucleon to hyperon spectra, and from
spherical to deformed nuclei. Open issues in this field are also discussed in
detail, including the perturbative nature, the supersymmetric representation
with similarity renormalization group, and the puzzle of intruder states.
