  This paper aims to deepen the theory of bijective non-degenerate
set-theoretic solutions of the Yang-Baxter equation, not necessarily
involutive, by means of q-cycle sets. We entirely focus on the finite
indecomposable ones among which we especially study two classes of current
interest: the simple solutions and those having finite primitive level. In
particular, we provide two group-theoretic characterizations of these
solutions, involving their permutation groups. Finally, we deal with some open
questions.
