  The open question of where, when, and how the heavy elements beyond iron
enrich our Universe has triggered a new era in nuclear physics studies.\ Of all
the relevant nuclear physics inputs, the mass of very neutron-rich nuclides is
a key quantity for revealing the origin of heavy elements beyond iron.\
Although the precise determination of this property is a great challenge,
enormous progress has been made in recent decades, and it has contributed
significantly to both nuclear structure and astrophysical nucleosynthesis
studies.\ In this review, we first survey our present knowledge of the nuclear
mass surface, emphasizing the importance of nuclear mass precision in
$r$-process calculations.\ We then discuss recent progress in various methods
of nuclear mass measurement with a few selected examples.\ For each method, we
focus on recent breakthroughs and discuss possible ways of improving the
weighing of $r$-process nuclides.
