  The inspiral of a stellar-mass compact object into a massive ($\sim
10^{4}$-$10^{7} M_{\odot}$) black hole produces an intricate gravitational-wave
signal. Due to the extreme-mass ratios involved, these systems complete $\sim
10^{4}$-$10^{5}$ orbits, most of them in the strong-field region of the massive
black hole, emitting in the frequency range $\sim10^{-4}-1~$Hz. This makes them
prime sources for the space-based observatory LISA (Laser Interferometer Space
Antenna). LISA observations will enable high-precision measurements of the
physical characteristics of these extreme-mass-ratio inspirals (EMRIs):
redshifted masses, massive black hole spin and orbital eccentricity can be
determined with fractional errors $\sim 10^{-4}$-$10^{-6}$, the luminosity
distance with better than $\sim 10\%$ precision, and the sky localization to
within a few square degrees. EMRIs will provide valuable information about
stellar dynamics in galactic nuclei, as well as precise data about massive
black hole populations, including the distribution of masses and spins. They
will enable percent-level measurements of the multipolar structure of massive
black holes, precisely testing the strong-gravity properties of their
spacetimes. EMRIs may also provide cosmographical data regarding the expansion
of the Universe if inferred source locations can be correlated with galaxy
catalogs.
