In this chapter, we study the structure monoid M(X, r) and the structure group G(X, r) of a solution (X, r) to the YBE. We prove that if (X, r) is involutive, then M(X, r) is cancellative. We show that G(X, r) has a natural structure of skew brace. Then we use the bijective correspondence between braided groups and skew braces, which we study in the first section of this chapter, to prove that the solution (X, r) induces a natural solution \((G(X,r),r_G)\) to the YBE such that \(r_G\) also is a braiding operator on G(X, r) with a nice universal property.