  We develop a theory of symmetry in open quantum systems. Using the
operator-state mapping, we characterize symmetry of Liouvillian superoperators
for the open quantum dynamics by symmetry of operators in the double Hilbert
space and apply the 38-fold internal-symmetry classification of non-Hermitian
operators. We find rich symmetry classification due to the interplay between
symmetry in the corresponding closed quantum systems and symmetry inherent in
the construction of the Liouvillian superoperators. As an illustrative example
of open quantum bosonic systems, we study symmetry classes of dissipative
quantum spin models. For open quantum fermionic systems, we develop the
$\mathbb{Z}_4$ classification of fermion parity symmetry and antiunitary
symmetry in the double Hilbert space, which contrasts with the $\mathbb{Z}_8$
classification in closed quantum systems. We also develop the symmetry
classification of open quantum fermionic many-body systems -- a dissipative
generalization of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad
master equation. We establish the periodic tables of the SYK Lindbladians and
elucidate the difference from the SYK Hamiltonians. Furthermore, from extensive
numerical calculations, we study its complex-spectral statistics and
demonstrate dissipative quantum chaos enriched by symmetry.
