  Nuclear weak decays provide important probes to fundamental symmetries in
nature. A precise description of these processes in atomic nuclei requires
comprehensive knowledge on both the strong and weak interactions in the nuclear
medium and on the dynamics of quantum many-body systems. In particular, an
observation of the hypothetical double beta decay without emission of neutrinos
($0\nu\beta\beta$) would unambiguously demonstrate the Majorana nature of
neutrinos and the existence of the lepton-number-violation process. It would
also provide unique information on the ordering and absolute scale of neutrino
masses. The next-generation tonne-scale experiments with sensitivity up to
$10^{28}$ years after a few years of running will probably provide a definite
answer to these fundamental questions based on our current knowledge on the
nuclear matrix element (NME), the precise determination of which is a challenge
to nuclear theory. Beyond-mean-field approaches have been frequently adapted
for the study of nuclear structure and decay throughout the nuclear chart for
several decades. In this review, we summarize the status of beyond-mean-field
calculations of the NMEs of $0\nu\beta\beta$ decay assuming the standard
mechanism of an exchange of light Majorana neutrinos. The challenges and
prospects in the extension and application of beyond-mean-field approaches for
$0\nu\beta\beta$ decay are discussed.
