  Hydrogen Lyman-$\alpha$ (Ly$\alpha$) emission has been one of the major
observational probes for the high redshift universe, since the first
discoveries of high-$z$ Ly$\alpha$ emitting galaxies in the late 1990s. Due to
the strong Ly$\alpha$ emission originated by resonant scattering and
recombination of the most-abundant element, Ly$\alpha$ observations witness not
only HII regions of star formation and AGN but also diffuse HI gas in the
circum-galactic medium (CGM) and the inter-galactic medium (IGM). Here we
review Ly$\alpha$ sources, and present theoretical interpretations reached to
date. We conclude that: 1) A typical Ly$\alpha$ emitter (LAE) at $z\gtrsim 2$
with a $L^*$ Ly$\alpha$ luminosity is a high-$z$ counterpart of a local dwarf
galaxy, a compact metal-poor star-forming galaxy (SFG) with an approximate
stellar (halo) mass and star-formation rate of $10^{8-9} M_\odot$ ($10^{10-11}
M_\odot$) and $1-10 M_\odot$ yr$^{-1}$, respectively; 2) High-$z$ SFGs
ubiquitously have a diffuse Ly$\alpha$ emitting halo in the CGM extending to
the halo virial radius and beyond; 3) Remaining neutral hydrogen at the epoch
of reionization makes a strong dimming of Ly$\alpha$ emission for galaxies at
$z>6$ that suggest the late reionization history. The next generation large
telescope projects will combine Ly$\alpha$ emission data with HI Ly$\alpha$
absorptions and 21cm radio data that map out the majority of hydrogen (HI+HII)
gas, uncovering the exchanges of i) matter by outflow/inflow and ii) radiation,
relevant to cosmic reionization, between galaxies and the CGM/IGM.
