Competition of the shell closure and deformations across the doubly magic $^{78}$Ni

The properties of the neutron-rich isotope $^{78}$Ni, long postulated to be doubly magic, have been extensively explored through recent experimental and theoretical studies. Confirmations of robust shell closures at $Z=28$ and $N=50$ as well as hints of competing deformations in neighboring isotopes have been obtained. Innovations of a thick liquid hydrogen target system with vertex reconstructions and the in-beam $\gamma$-ray spectroscopy technique have facilitated detailed investigations into the nuclear structure of these extreme systems. Proton knockout reactions conducted at relativistic energies have provided the first experimental evidence of shape coexistence at the cornerstone nucleus $^{78}$Ni and its vicinity. As the nuclear structure around $^{78}$Ni influences the description of very neutron-rich systems and r-process nucleosynthesis, these findings underscore the importance of further investigations. This review encapsulates the recent results concerning the nuclear structure at the vicinity of $^{78}$Ni on both experimental and theoretical aspects. It outlines prospective research directions that could further illuminate this complex and intriguing area of the nuclear chart.