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 $zgtrsim 2$ 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.
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