  This review summarizes our current understanding of the outer heliosphere and
local interstellar medium (LISM) inferred from observations and modeling of
interplanetary Lyman-$\alpha$ emission. The emission is produced by solar
Lyman-alpha photons (121.567 nm) backscattered by interstellar H atoms
inflowing to the heliosphere from the LISM. Studies of Lyman-alpha radiation
determined the parameters of interstellar hydrogen within a few astronomical
units from the Sun. The interstellar hydrogen atoms appeared to be decelerated,
heated, and shifted compared to the helium atoms. The detected deceleration and
heating proved the existence of secondary hydrogen atoms created near the
heliopause. This finding supports the discovery of a Hydrogen Wall beyond the
heliosphere consisting of heated hydrogen observed in HST/GHRS Lyman-alpha
absorption spectra toward nearby stars. The shift of the interstellar hydrogen
bulk velocity was the first observational evidence of the global heliosphere
asymmetry confirmed later by Voyager in situ measurements. SOHO/SWAN all-sky
maps of the backscattered Lyman-alpha intensity identified variations of the
solar wind mass flux with heliolatitude and time. In particular, two maxima at
mid-latitudes were discovered during solar activity maximum, which Ulysses
missed due to its specific trajectory. Finally, Voyager/UVS and New
Horizons/Alice UV spectrographs discovered extra-heliospheric Lyman-alpha
emission. We review these scientific breakthroughs, outline open science
questions, and discuss potential future heliospheric Lyman-alpha experiments.
