  Braces are generalizations of radical rings, introduced by Rump to study
involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation
(YBE). Skew braces were also recently introduced as a tool to study not
necessarily involutive solutions. Roughly speaking, skew braces provide
group-theoretical and ring-theoretical methods to understand solutions of the
YBE. It turns out that skew braces appear in many different contexts, such as
near-rings, matched pairs of groups, triply factorized groups, bijective
1-cocycles and Hopf-Galois extensions. These connections and some of their
consequences are explored in this paper. We produce several new families of
solutions related in many different ways with rings, near-rings and groups. We
also study the solutions of the YBE that skew braces naturally produce. We
prove, for example, that the order of the canonical solution associated with a
finite skew brace is even: it is two times the exponent of the additive group
modulo its center.
