A strongly-coupled plasma is a collection of free charged particles that interact with a Coulomb repulsion that is so strong that nearby particles do not easily move past one another. Unlike weakly-coupled plasmas, strongly-coupled plasmas exhibit a self-organization of particles into an arrangement like a solid crystalline lattice or a liquid. Dusty plasmas consist of micron-size particles of solid matter that are immersed in a plasma of electrons and ions. The dust particles gain a large electric charge and become strongly coupled. The motion of discrete particles can be tracked using a video microscopy diagnostic. Dusty plasma experiments allow a study of strongly-coupled plasma physics and an experimental simulation of condensed matter physics. Experiments are reported using a single layer of particles in the plasma to study two-dimensional (2D) physics. It is demonstrated experimentally that in addition to the solid and liquid states, a strongly-coupled dusty plasma can exist in an exotic state called a superheated solid. A 2D dusty plasma, initially self-organized in a crystalline lattice, is heated rapidly by rastered laser beams. The suspension remains in a solid lattice at a temperature well above the melting point. Shear-induced melting is studied in a 2D dusty plasma by applying shear to a crystalline lattice using a pair of oppositely-directed laser beams. Unexpectedly, coherent longitudinal waves are also excited in the resulting shear flow. In the first experiment of its kind, a suddenly-applied shear is found to produce a melting front that spreads at the transverse sound speed. The viscoelasticity of strongly-coupled plasmas in a liquid state is quantified. In the first experiment for any kind of physical system, the wavenumber-dependent viscosity, η(k), is computed from measurements of the random motion of particles. It is found that η(k) diminishes with increasing k, indicating that viscous behavior is gradually replaced by elastic behavior as the scale length is reduced. As a tool for studying transport at a microscopic level, the self-intermediate scattering function (self-ISF) is used in numerical simulations of 2D dusty plasmas. Two physical processes are studied using the self-ISF: relaxation of random motion, and melting. The wavenumber-dependence of the relaxation time in a liquid-phase strongly-coupled plasma is shown to be useful for distinguishing normal and anomalous diffusion. The self-ISF is also demonstrated to be a sensitive indicator of the melting transition. An improved image-analysis method is developed for calculating particle positions with minimal measurement errors. This development also provides an understanding of sources of error and the dependence on parameters that the experimenter can control.