  Stellar-mass black holes and neutron stars represent extremes in gravity,
density, and magnetic fields. They therefore serve as key objects in the study
of multiple frontiers of physics. In addition, their origin (mainly in
core-collapse supernovae) and evolution (via accretion or, for neutron stars,
magnetic spindown and reconfiguration) touch upon multiple open issues in
astrophysics.
  In this review, we discuss current mass and spin measurements and their
reliability for neutron stars and stellar-mass black holes, as well as the
overall importance of spins and masses for compact object astrophysics. Current
masses are obtained primarily through electromagnetic observations of binaries,
although future microlensing observations promise to enhance our understanding
substantially. The spins of neutron stars are straightforward to measure for
pulsars, but the birth spins of neutron stars are more difficult to determine.
In contrast, even the current spins of stellar-mass black holes are challenging
to measure. As we discuss, major inroads have been made in black hole spin
estimates via analysis of iron lines and continuum emission, with reasonable
agreement when both types of estimate are possible for individual objects, and
future X-ray polarization measurements may provide additional independent
information. We conclude by exploring the exciting prospects for mass and spin
measurements from future gravitational wave detections, which are expected to
revolutionize our understanding of strong gravity and compact objects.
