Set-theoretic solutions of the Yang-Baxter equation, braces and braided groups

Set-theoretic solutions of the Yang--Baxter equation form a meeting ground of mathematical physics, algebra and combinatorics.
Such a solution consists of a set X and a bijective map  $r:X\times X\to X\times X$ which satisfies the braid relations.
In this talk we shall discuss the close relation between set-theoretic solutions of YBE, braided groups, and braces.
We apply the theory of matched pairs for effective computation in braces. We determine explicitly all symmetric sets (X,r)  
for which the associated brace G(X,r) is a two-sided brace, or  inherits  other important combinatorial properties  from the solution (X,r).
We find new evidences confirming our conjecture that every finite square free-solution of YBE  is a multipermutation solution.