The mid- through far-infrared wavelength regime accounts for the second largest component of the spectral energy distribution of the Universe -- below the primordial cosmic background and similar in magnitude to the optical background. The far-infrared peak is due to prolific star formation within galaxies during the early lifetime of the Universe, with the stellar emission obscured by enshrouding dust and re-emitted at longer (as well as red-shifted) wavelengths. Despite their importance for the energy budget of the Universe, these infrared wavelengths remain among the least investigated, primarily due to the difficulty in launching appropriate instruments. Beyond the clear importance of the mid- through far-infrared continuum as an indicator of how, where, and when the galaxies in the Universe emit energy, this wavelength regime is also abundant in spectroscopic diagnostics, including those from light molecules such as H_2 and HD, important for direct determinations of gas mass (independent of metallicity and freeze-out). This is also the most important part of the electromagnetic spectrum for observations of water, in both gas and ice form, and thus a key probe into the pathway to life around stars. Furthermore, a wealth of fine-structure lines probes the ionized regions around hot stars and active galactic nuclei -- providing essential diagnostics of temperature, metallicity, and hardness of the radiation field. Similarly, dust features manifest strongly in this window, including emission from PAHs, silicates, minerals, and both crystalline and amorphous ices. When combined with the fine-structure lines, PAH and dust emission provide important diagnostics for the energy budget of galaxies. Many of the specific science questions posed by mid- through far-infrared observations, and discussed in this white paper, directly relate to how the Universe, our Galaxy, our Sun, our Solar System, and the Earth began and have evolved up to this point in time (indeed that is why the NASA-concept is named the Origins Space Telescope). In order to provide answers to these questions, the above studies necessarily bring together the entire "astro-turf", including astronomy, astrophysics, astrochemistry, and astrobiology. The next generation of mid- through far-infrared space telescopes are already in the planning stage (i.e. SPICA and Origins). While these telescopes will not improve significantly the angular resolution over previous missions, they are both designed to be extremely sensitive, beating Herschel by two to three orders of magnitude. The science cases utilizing such observational opportunities discussed in this white paper show both the importance of these wavelengths for a broad range of research and the leadership roles of Canadian astronomers in these fields. It is also important to note that, as was the case with previous mid- through far-infrared missions, this extreme increase in telescope capability will almost certainly lead to significant `discovery science', unanticipated at present.

White paper identifier W035