  Braces were introduced by Rump to study non-degenerate involutive
set-theoretic solutions of the Yang-Baxter equation. We generalize Rump's
braces to the non-commutative setting and use this new structure to study not
necessarily involutive non-degenerate set-theoretical solutions of the
Yang-Baxter equation. Based on results of Bachiller and Catino and Rizzo, we
develop an algorithm to enumerate and construct classical and non-classical
braces of small size up to isomorphism. This algorithm is used to produce a
database of braces of small size. The paper contains several open problems,
questions and conjectures.
