Terahertz waves have great potentials for security screening, medical imaging, and chemical identification. The use of plasmonic contact electrodes in photoconductive terahertz emitters has been very effective in enhancing terahertz radiation power and signal-to-noise ratio levels of terahertz imaging and sensing systems. However, previously demonstrated plasmonic photoconductive terahertz emitters have all utilized grating-based plasmonic contact electrodes, which are sensitive to the polarization state of their optical pump beam. This polarization sensitivity can degrade the performance of plasmonic photoconductive terahertz emitters in practical application settings. In this thesis, we develop a polarization-insensitive plasmonic photoconductive terahertz emitter, which utilizes a periodic array of subwavelength cross-shaped apertures as the plasmonic contact electrodes. By using cross-shaped aperture arrays, surface plasmon waves can be excited near the metal contact-substrate interface, bringing more photo-generated carriers close to this interface, which reduces the carrier transport path length to the contact electrodes. The two-dimensional symmetry of the cross-shaped apertures leads to a polarization-insensitive interaction between the plasmonic contact electrodes and optical pump beam. The geometry of the cross-shaped aperture arrays is selected to achieve maximum optical pump absorption in the photo-absorbing substrate at the metal-contact interface. Prototypes of this plasmonic photoconductive emitter are fabricated. The fabricated emitter prototypes show an excellent polarization insensitivity to the optical pump. The terahertz radiation power and efficiency of this emitter are comparable to those of a grating-based plasmonic photoconductive emitter with the same terahertz radiating elements.