The Beckman-Quarles theorem states that every unit-distance preserving mapping \(f : \mathbb {E}^n \to \mathbb {E}^n\) is an isometry if n is an integer greater than 1. Section 8.1 is devoted to the discussion of whether the Beckman-Quarles theorem also holds in rational n-spaces. It is known that every unit-distance preserving mapping \(f : \mathbb {Q}^n \to \mathbb {Q}^n\) is an isometry if n is an even integer greater than 5 or 5 or an odd integer of the form \(n = 2m^2 - 1\) , where \(m > 2\) . We have to omit the interesting proofs of all theorems introduced in this section due to space constraints. In Sect. 8.2, we will discuss the theory of tensegrity structures that F. Rádo et al. used to partially solve the Aleksandrov-Rassias problems. Most of the content in this section comes from the paper by Bezdek and Connelly (Period Math Hungar 39(1–3):185–200, 1999). Indeed, they were able to improve the result of Rádo et al. even further by refining the idea presented by Rádo et al. In Sects. 8.3 and 8.4, we provide some sufficient conditions for the Benz-Berens theorem and the Beckman-Quarles theorem to also hold in an open convex set. The contents of those sections are mainly based on the papers by Jung (Bull Braz Math Soc (NS) 37(3):351–359, 2006); Jung (Bull Braz Math Soc (NS)40(1):77–84, 2009); Jung and Rassias (J Korean Math Soc 41(4):667–680, 2004). In the final section, we assume that the Beckman-Quarles theorem does not assume that the mapping preserves a certain distance but rather a certain geometric figure. S.-M. Jung and B. Kim have achieved interesting results on this topic, which we will systematically present in the last section.