A commutative and unital ring \(R\) is an AV-ring (Almost Valuation Ring) if for every \(a\), \(b\) in \(R\) there exists a positive integer \(n\) such that \(a^nR\subseteq b^nR\) or \(b^nR\subseteq a^nR\). Being an AV-ring implies being quasi-local. If \(R\) is a domain, then it is an AV-domain if for every \(a\) in its quotient field there is a positive integer \(n\) such that \(a^n\in R\) or \(a^{-n}\in R\). Now, let \(S\) be a commutative overring of \(R\) with the same unit element. Then \((R,S)\) is an AV-ring pair if for every ring in the set \([R,S]\) (of all rings \(T\) such that \(R\subseteq T\subseteq S\)) is an AV-ring. In the same way, \((R,S)\) is an AV-domain pair if every element of \([R,S]\) is an AV-domain. This paper provides a broad study of the AV-ring pairs. The reader can find more or less thirty necessary and sufficient conditions (NSC) for being an AV-ring pair, ten for being an AV-domain pair, and other characterizations. For example, \((R,S)\) is an AV-ring pair if, and only if, \((R,\overline{R}_S)\) and \((\overline{R}_S,S)\) are AV-ring pairs, where \(\overline{R}_S\) is the integral closure of \(R\) in \(S\). The authors look at extensions of rings \((R\subset S)\) satisfying different conditions, and they get NSC for \((R,S)\) being an AV-ring pair. For example, if \(R\) is a field and \(S\) is a nonzero \(R\)-algebra, then a NSC is: \(R\subset S\) being an integral extension and \(S\) being quasi-local. They consider reduced rings (i.e.they do not contain nonzero nilpotent elements), normal pairs (i.e.every element of \([R,S]\) is integrally closed in \(S\)), P-extensions (every element of \(S\) is the root of a polynomial at least one of whose coefficients is a unit), root extensions (for every \(s\in S\) there is a positive integer \(n\) such that \(s^n\in R\)). Several propositions are dedicated to inductive families of AV-ring pairs, and we can deduce NSC for being AV-ring pairs which rely on the finite type \(R\) sub-algebras of \(S\). The authors show that being an AV-ring or an AV-ring pair can be proved by means of pullback of quotients by prime ideals. A section is dedicated to the case where \((R\subset S)\) is a minimal pair (i.e.\([R,S]=\{R,S\}\)). The sixth section focuses on the case where \(R\) is not an AV-domain, but every ring in \(]R,S]\) is an AV-domain. In the last section, locally divided domains are studied (if \(R\) is a domain, then it is locally divided if every maximal ideal \(M\) is equal to \(MR_M\)). In this section, some proofs rely on going-down (\(R\subset S\) satisfies going-down if for every chain \(I_1\supseteq \cdots \supseteq I_n\) of prime ideals of \(R\) such that there exists a chain \(J_1\supseteq \cdots \supseteq J_{n-1}\) of prime ideals of \(S\) such that every \(J_k\) lies over \(I_k\) (\(1\leq k<n\)), the later chain can be completed in a chain \(J_1\supseteq \cdots \supseteq J_n\) where \(J_n\) lies over \(I_n\)). So, many aspects of AV-ring pairs and AV-domain pairs are studied through the eight lemmas, eleven propositions, nine theorems, fifteen corollaries, three examples and three remarks of this paper.