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This review summarizes our current understanding of the outer heliosphere and local interstellar medium (LISM) inferred from observations and modeling of interplanetary Lyman-$α$ 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.