Solid-state polymer electrolytes and high-concentration liquid electrolytes, such as water-in-salt electrolytes and ionic liquids, are emerging materials to replace the flammable organic electrolytes widely used in industrial lithium-ion batteries. Extensive efforts have been made to understand the ion transport mechanisms and optimize the ion transport properties. This perspective reviews the current understanding of the ion transport and polymer dynamics in liquid and polymer electrolytes, comparing the similarities and differences in the two types of electrolytes. Combining recent experimental and theoretical findings, we attempt to connect and explain ion transport mechanisms in different types of small-molecule and polymer electrolytes from a theoretical perspective, linking the macroscopic transport coefficients to the microscopic, molecular properties such as the solvation environment of the ions, salt concentration, solvent/polymer molecular weight, ion pairing, and correlated ion motion. We emphasize universal features in the ion transport and polymer dynamics by highlighting the relevant time and length scales. Several outstanding questions and anticipated developments for electrolyte design are discussed, including the negative transference number, control of ion transport through precision synthesis, and development of predictive multiscale modeling approaches.