  Strong octupole correlations have been observed in the low-lying states of
atomic nuclei across various mass regions. In this review, we provide an
overview of Beyond Mean-Field (BMF) studies of nuclear octupole collective
motions with Generator Coordinate Method (GCM) in combination with
quantum-number projections that are implemented to restore the broken
symmetries in nuclear mean-field states. We highlight recent developments
within this framework and their applications to excitation spectra and
electromagnetic transition rates in octupole-shaped nuclei and hypernuclei. We
discuss the novel phenomena of nucleon clustering in light nuclei.
Additionally, we explore the phase transition from octupole vibrations to
rotational motions as spin increases in heavy nuclei. Lastly, we examine the
status and future prospects of studies on octupole deformation effects in
nuclear Schiff moments. These studies, along with the upper limits of atomic
Electric Dipole Moment (EDM), impose stringent constraints on
beyond-standard-model time-reversal-violating nucleon-nucleon interactions.
