  The ordering of the neutrino masses is a crucial input for a deep
understanding of flavor physics, and its determination may provide the key to
establish the relationship among the lepton masses and mixings and their
analogous properties in the quark sector. The extraction of the neutrino mass
ordering is a data-driven field expected to evolve very rapidly in the next
decade. In this review, we both analyze the present status and describe the
physics of subsequent prospects. Firstly, the different current available tools
to measure the neutrino mass ordering are described. Namely, reactor,
long-baseline (accelerator and atmospheric) neutrino beams, laboratory searches
for beta and neutrinoless double beta decays and observations of the cosmic
background radiation and the large scale structure of the universe are
carefully reviewed. Secondly, the results from an up-to-date comprehensive
global fit are reported: the Bayesian analysis to the 2018 publicly available
oscillation and cosmological data sets provides \emph{strong} evidence for the
normal neutrino mass ordering versus the inverted scenario, with a significance
of 3.5 standard deviations. This preference for the normal neutrino mass
ordering is mostly due to neutrino oscillation measurements. Finally, we shall
also emphasize the future perspectives for unveiling the neutrino mass
ordering. In this regard, apart from describing the expectations from the
aforementioned probes, we also focus on those arising from alternative and
novel methods, as 21~cm cosmology, core-collapse supernova neutrinos and the
direct detection of relic neutrinos.
