  We produce novel non-involutive solutions of the Yang-Baxter equation coming
from (skew) braces. These solutions are generalisations of the known ones
coming from braces and skew braces, and surprisingly in the case of braces they
are not necessarily involutive. In the case of two-sided (skew) braces one can
assign such solutions to every element of the set. Novel bijective maps
associated to the inverse solutions are also introduced. Moreover, we show that
the recently derived Drinfeld twists of the involutive case are still
admissible in the non-involutive frame and we identify the twisted $r$-matrices
and twisted coproducts. We observe, as in the involutive case that the
underlying quantum algebra is not a quasi-triangular bialgebra, as one would
expect, but a quasi-triangular quasi-bialgebra. The same applies to the quantum
algebra of the twisted $r$-matrices, contrary to the involutive case.
