  We perform a systematic symmetry classification of many-body Lindblad
superoperators describing general (interacting) open quantum systems coupled to
a Markovian environment. Our classification is based on the behavior of the
many-body Lindbladian under antiunitary symmetries and unitary involutions. We
find that Hermiticity preservation reduces the number of symmetry classes,
while trace preservation and complete positivity do not, and that the set of
admissible classes depends on the presence of additional unitary symmetries: in
their absence or in symmetry sectors containing steady states, many-body
Lindbladians belong to one of ten non-Hermitian symmetry classes; if however,
there are additional symmetries and we consider non-steady-state sectors, they
belong to a different set of 19 classes. In both cases, it does not include
classes with Kramer's degeneracy. Remarkably, our classification admits a
straightforward generalization to the case of non-Markovian, and even
non-trace-preserving, open quantum dynamics. While the abstract classification
is completely general, we then apply it to general (long-range, interacting,
spatially inhomogeneous) spin-$1/2$ chains. We explicitly build examples in all
ten classes of Lindbladians in steady-state sectors, describing standard
physical processes such as dephasing, spin injection and absorption, and
incoherent hopping, thus illustrating the relevance of our classification for
practical physics applications. Finally, we show that the examples in each
class display unique random-matrix correlations. To fully resolve all
symmetries, we employ the combined analysis of bulk complex spacing ratios and
the overlap of eigenvector pairs related by symmetry operations. We further
find that statistics of levels constrained onto the real and imaginary axes or
close to the origin are not universal due to spontaneous breaking of
Lindbladian PT symmetry.
