Metric fixed-point theory is one of the most extensively researched mathematics subjects. We express and prove fixed-point theorems using various approaches, both on the structure of metric spaces and on generalized metric space structures. Along with the help of the hybrid-interpolative Reich–Istratescu-type \(S-{a}\mu \) )-contraction structure, new fixed-point results are found on S-metric spaces in this chapter. Every new result obtained sheds light on another research topic. These results significantly advance fixed-point theory by generalizing existing results in the literature.