Mass loss from asymptotic giant branch (AGB) stars creates an envelope that fosters the synthesis of a variety of both simple and complex molecules. AGB stars evolve into planetary nebula (PN), and the star advances into the white dwarf stage with an increasing flux of ultraviolet (UV) photons. Chemical models have predicted that the molecular species created in the AGB phase are destroyed by the strong UV radiation fields generated by the white dwarf. To test this hypothesis, millimeter- and submillimeter-wave observations of a variety of PNe of varying morphologies and kinematic ages have been conducted using the 12-m and Submillimeter Telescopes of the Arizona Radio Observatory (ARO) and the 30-m Telescope of the Institut de Radioatronomie Millimètrique (IRAM). In contrast to model predictions, searches for HCN and HCO+ in seventeen PNe using the ARO facilities resulted in detection in ~76% of the sample objects, while subsequent measurements of HNC and CCH in the 11 PNe in which HCN had been identified yielded 100% and ~82% detections, respectively. The results of these observations strongly indicate that polyatomic molecular material is a common constituent of PN ejecta. Further, the abundances of these species do not vary significantly with nebular age, in sharp contrast to the predictions of chemical models. In addition, the observed abundances are consistently 1-2 orders of magnitude greater than those measured for diffuse clouds, supporting the argument that these molecules are shielded from the strong UV radiation of the central star within dense clumps of gas and dust which slowly disperse and seed the surrounding interstellar medium (ISM) with molecular material. These conclusions are strengthened by the identification of HCN, HNC, CCH, and c-C3H2 at eight positions across the very old Helix Nebula. Of particular interest are observations of 16 different chemical species, including rare isotopic variants, in the young PN K4-47 using both the ARO facilities and the IRAM 30-m. Some of these species have never before been identified in a PN, such as CH3CN, H2CNH, and CH3CCH. Further, numerous 13C-containing species, including all singly- and doubly-13C substituted variants of HC3N, were detected in the source. Most unusual are the astoundingly low 12C/13C, 14N/15N, and 16O/17O ratios measured in the PN (2.2 ± 0.8, 13.6 ± 6.5, and 21.4 ± 10.3, respectively); the latter two are the lowest thus far measured in interstellar gas. These results have important implications on stellar nucleosynthesis and the sources of SiC presolar grains.