  We consider iterated function systems $\mathrm{IFS}(T_1,\dots,T_k)$
consisting of continuous self maps of a compact metric space $X$. We introduce
the subset $S_{\mathrm{t}}$ of {\emph{weakly hyperbolic sequences}}
$\xi=\xi_0\ldots\xi_n \ldots \in \Sigma_k^+$ having the property that
$\bigcap_n T_{\xi_{0}}\circ\cdots\circ T_{\xi_{n}}(X)$ is a point
$\{\pi(\xi)\}$. The target set $\pi(S_{\mathrm{t}})$ plays a role similar to
the semifractal introduced by Lasota-Myjak.
  Assuming that $S_{\mathrm{t}}\ne \emptyset$ (the only hyperbolic-like
condition we assume) we prove that the IFS has at most one strict attractor and
we state a sufficient condition guaranteeing that the strict attractor is the
closure of the target set. Our approach applies to a large class of genuinely
non-hyperbolic IFSs (e.g. with maps with expanding fixed points) and provides a
necessary and sufficient condition for the existence of a globally attracting
fixed point of the Barnsley-Hutchinson operator. We provide sufficient
conditions under which the disjunctive chaos game yields the target set (even
when it is not a strict attractor).
  We state a sufficient condition for the asymptotic stability of the Markov
operator of a recurrent IFS. For IFSs defined on $[0,1]$ we give a simple
condition for their asymptotic stability. In the particular case of IFSs with
probabilities satisfying a "locally injectivity" condition, we prove that if
the target set has at least two elements then the Markov operator is
asymptotically stable and its stationary measure is supported in the closure of
the target set.
