  The outskirts of galaxies offer extreme environments where we can test our
understanding of the formation, evolution, and destruction of molecules and
their relationship with star formation and galaxy evolution. We review the
basic equations that are used in normal environments to estimate physical
parameters like the molecular gas mass from CO line emission and dust continuum
emission. Then we discuss how those estimates may be affected when applied to
the outskirts, where the average gas density, metallicity, stellar radiation
field, and temperature may be lower. We focus on observations of molecular gas
in the outskirts of the Milky Way, extragalactic disk galaxies, early-type
galaxies, groups, and clusters. The scientific results show the versatility of
molecular gas, as it has been used to trace Milky Way spiral arms out to a
galactocentric radius of 15 kpc, to study star formation in extended
ultraviolet disk galaxies, to probe galaxy interactions in polar ring S0
galaxies, and to investigate ram pressure stripping in clusters. We highlight
the physical stimuli that accelerate the formation of molecular gas, including
internal processes such as spiral arm compression and external processes such
as interactions.
