In the Galaxy, microwave radiation can be amplified in the interstellar medium in the immediate neighborhood of young stellar objects, or circumstellar envelopes around evolved stars, resulting in cosmic maser emission. Cosmic masers exist because, in contrast to terrestrial conditions, the interstellar gas density is very low so that level population in molecules is typically not in thermal equilibrium, and sometimes inverted. In the nuclear regions of external galaxies, there exist very powerful OH ( 18 cm) and H2O ( 1.35 cm) cosmic masers with line luminosities of 102 104 L⊙, 106 times more luminous than typical Galactic maser sources. These are the "mega-masers," found in high-density molecular gas located within parsecs of active galactic nuclei in the case of H2O mega-masers, or within the central 100 pc of nuclear star-burst regions in the case of OH mega-masers. H2O mega-masers are most frequently found in galactic nuclei with Seyfert2 or LINER spectral characteristics, in spiral and some elliptical galaxies. OH mega-masers are found in ultra-luminous IR galaxies (ULIRG) with the warmest IR colors, and importantly, the OH luminosity is observed to increase with the IR luminosity: LOH L1.2IR. Because of the extremely high-surface brightness, H2O mega-maser emission can be mapped at sub-milli-arc-second resolution by Very Long Baseline Interferometry (VLBI), providing a powerful tool to probe spatial and kinematic distributions of molecular gas in distant galactic nuclei at scales below one parsec. An excellent example is the active galaxy, NGC 4258, in which mapping of the H2O mega-maser emission has provided the first direct evidence in an active galactic nucleus for the existence of a thin Keplerian accretion disk with turbulence, as well as highly compelling evidence for the existence of a massive black hole. The NGC 4258 mega-maser has also provided a geometric distance determination of extremely high precision. H2O mega-maser emission is also found to arise from postshocked gas from the impact of nuclear jets or outflows on the surrounding molecular clouds. High-resolution observations have shown that OH mega-masers originate from the molecular gas medium in 100-pc scale nuclear star-burst regions. It is proposed that such extreme star-burst regions, with extensive high-density gas bathed in a very high far-IR radiation field, are conducive to the formation of a very large number of OH maser sources that collectively produce the OH mega-maser emission. In the early Universe, galaxies or mergers could go through a very luminous phase, powered by intensive star-bursts and AGN formation, and could have extremely large OH and H2O maser luminosities, possibly producing giga-masers. With the increasing sensitivity of new telescopes and receivers, surveys and high-resolution studies of mega-masers and giga-masers will be very important tracers and high-resolution probes of active galactic nuclei, dust embedded star-bursts in the earliest galaxies and galaxy mergers in the epoch of very active star formation at z 2 and beyond. Distance determination of giga-masers at z 1 2 can provide on independent measure of how fast the universe is expanding.