In this thesis, we investigated the transmission characteristics of classical waves, both ultrasonic waves and microwaves, through subwavelength aperture arrays in metallic plates. We found that the transmissions of the waves at some frequencies are enhanced by up to ∼100% regardless of the area fraction of the aperture. The enhanced transmissions are attributed to two types of resonances that occur in the individual aperture and the periodic structure of the array. For aperture-related resonance, we showed its robustness to changes in the incidence angle and the localized distribution of the surface field by studying the microwave transmission of an array of fractal-shaped apertures. For structure-related resonance, the relevant characteristics, such as the angular dependence of the transmission and the modulation of the surface field in the region between the apertures, are opposite to those of aperture-related resonance, which are exhibited in the microwave transmission of an array of subwavelength holes. By investigating acoustic transmission through an array of small holes, we found that two types of resonances appear and dominate in different regions of a transmission spectrum. In fact, they are the limiting behaviors of a unified phenomenon. There is a smooth transition/link between the aperture-factor-induced and the structure-factor-induced transmissions in the acoustic case. In addition, we compared electromagnetic and the acoustic transmissions, and identified their similarities and differences in the aperture structures studied.