  Braces were introduced by Rump as a promising tool in the study of the
set-theoretic solutions of the Yang-Baxter equation. It has been recently
proved that, given a left brace $B$, one can construct explicitly all the
non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation
such that the associated permutation group is isomorphic, as a left brace, to
$B$. It is hence of fundamental importance to describe all simple objects in
the class of finite left braces. In this paper we focus on the matched product
decompositions of an arbitrary finite left brace. This is used to construct new
families of finite simple left braces.
