The uranyl ion is the predominant uranium species in environmental chemistry. In solution in natural waters, the uranyl ion encounters a variety of ligands, including organic ligands, especially carboxylates. The behavior of the uranyl ion is governed by the complexes it forms in solution. The most precise information available concerning uranyl complexation chemistry comes from solved and refined crystal structures. Hundreds of crystal structures of uranyl complexes with organic ligands have been published. Therefore, a system of structure symbols to classify uranyl-organic complexes is devised. The method was applied to describe 592 published structures of uranyl-organic complexes, then used to catalog the complex chemistry of uranyl carboxylates. A mild hydrothermal synthesis method produced crystals of novel uranyl oxalates. These compounds all display hydroxyl bridging between uranyl ions, leading to highly polymeric complexes. Syntheses attempting to create other uranyl carboxylates resulted in uranyl oxalates. These compounds were described using the structure symbol method. Data on dissolved uranyl-organic complexes was examined, and the complexes described were assigned structure symbols and compared with crystalline complexes. An attack on the question of a possible relationship between uranyl ion bond lengths or $u_3$ vibrational frequencies and the uranyl ion's equatorial coordination was explored, again using the structure symbol. Several conclusions may be drawn from this work. First, the uranyl ion may readily catalyze the oxidation of other carboxylic acids to oxalate at elevated temperatures, resulting in uranyl oxalate complexes in a variety of organic-rich environments. Second, the novel uranyl oxalates discovered are polymeric and probably less water-soluble than previously known uranyl oxalates. Third, the method of structure symbols has much potential to make the crystal chemistry of uranyl-organic complexes more organized and useful to chemists in a variety of fields.