Let \(S\) be a semigroup and \(a\in S\); the semigroup with underlying set \(S\) and multiplication \(\circ\) defined by \(x\circ y=xay\) is a variant of \(S\), denoted \((S,a)\). An element of a regular semigroup is regularity preserving if \((S,a)\) is regular. The paper is devoted to the study of the structure of the variants of a semigroup and the set of all regularity preserving elements of a regular semigroup. In particular, the set of all regularity preserving elements of a regular Rees matrix semigroup is studied. Concerning the structure of the variants, it is shown that all variants of a semigroup \(S\) are orthodox if and only if \(S\) is locally orthodox. Moreover, all variants of a regular semigroup are \(E\)-inversive and the set of all regular elements of each variant \((S,a)\) forms a subsemigroup of \((S,a)\).