Superiorization can be thought of, in some cases, as lying between feasibility-seeking and constrained minimization. It is not trying to solve the full fledged constrained minimization problem; rather, the task is to find a feasible point which is superior, with respect to an objective function value, to one returned by a feasibility-seeking only algorithm. We distinguish between weak superiorization and strong superiorization and clarify their nature. Superiorization has been successfully used in some important practical applications such as image reconstruction from projections, intensity-modulated radiation therapy and nondestructive testing, and awaits to be implemented and tested in additional fields.