In this talk, I will present several models of a few quantum particles interacting in one-dimension. These models are inspired by recent experiments with few-body and many-body ultracold atomic gases in tight optical waveguides. Such systems can have a rich and tunable variety of symmetries depending on the nature of the particles, the trap, and the interaction. When there is enough symmetry, few-body models can be solvable, integrable, or even superintegrable. Solvable models are touchstones for analysis and approximation, and they have a long history of shedding light on the dynamics and thermodynamics of interacting quantum systems. Further, symmetry analysis provides a method for classifying different models and relating them through symmetry breaking. One goal of this research is to exploit symmetry analysis to probe the relations among integrability and chaos, entanglement and coherence, and quantum thermalization. Another goal is to develop experimental protocols that use symmetry-protected control to generate states that are resources for quantum measurement and computation.