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.
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