The chemistry of gas and stars in galaxies connects many areas of astrophysics, from star and planet formation on small scales, to the enrichment and physical state of the intergalactic and circumgalactic media on large scales. The abundance patterns in nearby galaxies have been studied in great detail, but it is only in the last decade that we have been able to extend our knowledge of galaxy enrichment to the distant universe. I will discuss recent advances that have been made at Cosmic Noon (z~2-3, 10-12 Gyr ago) using large near-infrared spectroscopic galaxy surveys, as well as implications for studying galaxies during the Epoch of Reionization (z>6, less than 1 Gyr after the Big Bang) with facilities like the James Webb Space Telescope. One of the most interesting results has been the confirmation of significant and ubiquitous alpha-enhancement in high-redshift galaxies, which is a consequence of their rapid assembly histories. This chemical signature separates nascent galaxies in the early universe from the majority of galaxies today and explains many of the observed differences between these populations. I will conclude by presenting a new method of measuring the chemistry of high-redshift galaxies that explicitly accounts for these differences. These efforts are central to the success of forthcoming galaxy surveys and will provide the means to unify studies of galaxies across cosmic time.