Dusty plasmas, defined as plasmas of ions, electrons, neutrals, and charged micron to sub-micron dust particles, support a rich diversity of physical states. These states (ranging from solids to liquids to gas) are determined by the ratio of the Coulomb potential energy between dust particles to the particles kinetic energy and allow for a broad range of phenomena, from crystallization to dust acoustic waves. Due to various dusty plasma interactions, dust acoustic waves can be nonlinear and exhibit various wave phenomena, from topological wave defects to shock waves to structure formations. In this thesis, I investigate a spectrum of plasma and wave interactions in liquid-like dusty plasmas and focus on a range of dust acoustic wave phenomena observed experimentally in a dc discharge dusty plasma. By developing various experimental techniques, dust acoustic wave diffraction and topological wave defects, dust acoustic shock waves, temporal dust acoustic wave growth, and structure forming dust acoustic waves were observed. I begin in Chapter 2 with the diffraction of dust acoustic waves, which I investigated by introducing a glass rod into the dusty plasma. The resulting diffraction pattern was compared to acoustic wave diffraction in a neutral gas. In addition to the diffraction pattern, topological wave defects were observed to form. I continue Chapter 2 with a preliminary investigation into topological wave defects in dust acoustic waves. Chapter 3 follows with three nonlinear dust acoustic wave experiments. I created a shock tube like profile for dust acoustic waves using a single slit. The shock-tube like potential resulted in two sets of nonlinear dust acoustic waves, coalescing high and low amplitude waves and dust acoustic waves that developed into dust acoustic shock waves. The self-excited dust acoustic shock waves were compared to theoretical models. The third nonlinear dust acoustic wave phenomenon that I investigated was a reverse drift mode that appears in high amplitude dust acoustic waves. I propose a wave process based on dust particle dynamics in high amplitude dust acoustic waves to explain the observations. In Chapter 4, I describe an experimental technique that I developed to create a quiescent drifting dusty plasma. The drifting dusty plasma was used to observe dust acoustic wave growth from thermal density fluctuations. The observed growth rate and frequency were compared kinetic and fluid models. In Chapter 5, I describe experimental observations of a structure forming instability in dusty plasmas. By increasing the discharge current, transient and aperiodic dust density striations formed. I characterized the transient and stationary modes and compared the stationary mode to an ionization/ion-drag instability and a polarization instability.