The research described in this dissertation is part of an ongoing investigation of student learning of the wave-like properties of light and matter. The investigation was conducted in the context of a broad range of physics courses, including courses for introductory students, advanced undergraduate majors, and graduate students. The initial emphasis was to strengthen the current research base on student understanding of physical optics. The scope of the investigation was later expanded to probe the ability of students to interpret the interference and diffraction of matter in terms of a wave model. The findings led us to extend the investigation further to include student understanding of the de Broglie wavelength and some very basic concepts in quantum mechanics. Additional difficulties in interpreting formal representations were found to cause confusion about the nature of light as an electromagnetic wave and inhibit the ability of students to predict the outcome of simple quantum mechanical measurements. The results from the investigation were used to guide the design of supplementary instructional materials that have been shown to address specific difficulties identified in the research.