The detection and characterization of two classes of ultralong-range Rydberg molecules known as "trilobite" and "butterfly" molecules is presented. These molecules are a subset of a class of Rydberg molecules which asymptotically consist of a Rydberg atom and a ground state atom. The trilobite and butterfly molecules have giant, body-fixed permanent dipole moments on the order of 1000 Debye. The two classes of molecules are distinguished by the relative dominance of the s-wave and p-wave electron scattering. Spectra for (nS_1/2 + 6S_1/2) ^3 Sigma molecules, where n = 37, 39 and 40 and measurements of the Stark broadenings of selected trilobite states in Cs due to the application of a constant external electric field are presented. Additionally, measurements of spectra and Stark splittings for p-wave dominated (nS_1/2 + 6S_1/2) ^3 Pi molecules, where n = 31 and 32 are presented. Computational work on Rydberg pair interactions is also discussed.