This dissertation describes experiments which explore the structure and dynamics in two classes of soft materials: lyotropic chromonic liquid crystals and colloidal glasses and super-cooled liquids. The first experiments found that the achiral LCLCs, sunset yellow FCF (SSY) and disodium cromoglycate (DSCG) both exhibit spontaneous mirror symmetry breaking in the nematic phase driven by a giant elastic anisotropy of their twist modulus compared to their splay and bend moduli. Resulting structures of the confined LCLCs display interesting director configurations due to interplay of topologically required defects and twisted director fields. At higher concentrations, the LCLC compounds form columnar phases. We studied the columnar phase confined within spherical drops and discovered and understood configurations of the LC that sometimes led to non-spherical droplet shapes. The second experiments with SSY LCLCs confined in hollow cylinders uncovered director configurations which were driven in large measure by an exotic elastic modulus known as saddle-splay. We measured this saddle-splay modulus in a LCLC for the first time and found it to be more than 50 times greater than the twist elastic modulus. This large relative value of the saddle-splay modulus violates a theoretical result/assumption known as the Ericksen inequality. A third group of experiments on LCLCs explored the drying process of sessile drops containing SSY solutions, including evaporation dynamics, morphology, and deposition patterns. These drops differ from typical, well-studied evaporating colloidal drops primarily due to the LCLC's concentration-dependent isotropic, nematic, and columnar phases. Phase separation occurs during evaporation, creating surface tension gradients and significant density and viscosity variation within the droplet. Thus, the drying multiphase drops exhibit new convective currents, drop morphologies, deposition patterns, as well as a novel ordered crystalline phase. Finally, experiments in colloidal glasses and super-cooled liquids were initiated to probe the relationship between structure and dynamics in their constituent particles. The displacements of individual particles in the colloids can be decomposed into small cage fluctuations and large rearrangements into new cages. We found a correlation between the rate of rearrangement and the local cage structure associated with each particle. Particle trajectories of a two-dimensional binary mixture of soft colloids are captured by video microscopy. We use a machine learning method to calculate particle ``softness'', which indicates the likelihood of rearrangement based on many radial structural features for each particle. We measured the residence time between consecutive rearrangements and related probability distribution functions (PDFs). The softness-dependent conditional PDF is well fit by an exponential with decay time decreasing monotonically with increasing softness. Using these data and a simple thermal activation model, we determined activation energies for rearrangements.