  We investigate different notions of recognizability for a free monoid
morphism $\sigma: \mathcal{A}^* \to \mathcal{B}^*$. Full recognizability occurs
when each (aperiodic) point in $\mathcal{B}^\mathbb{Z}$ admits at most one
tiling with words $\sigma(a)$, $a \in \mathcal{A}$. This is stronger than the
classical notion of recognizability of a substitution $\sigma:
\mathcal{A}^*\to\mathcal{A}^*$, where the tiling must be compatible with the
language of the substitution. We show that if $|\mathcal A|=2$, or if
$\sigma$'s incidence matrix has rank $|\mathcal A|$, or if $\sigma$ is
permutative, then $\sigma$ is fully recognizable. Next we investigate the
classical notion of recognizability and improve earlier results of Moss\'{e}
(1992) and Bezuglyi, Kwiatkowski and Medynets (2009), by showing that any
substitution is recognizable for aperiodic points in its substitutive shift.
Finally we define recognizability and also eventual recognizability for
sequences of morphisms which define an $S$-adic shift. We prove that a sequence
of morphisms on alphabets of bounded size, such that compositions of
consecutive morphisms are growing on all letters, is eventually recognizable
for aperiodic points. We provide examples of eventually recognizable, but not
recognizable, sequences of morphisms, and sequences of morphisms which are not
eventually recognizable. As an application, for a recognizable sequence of
morphisms, we obtain an almost everywhere bijective correspondence between the
$S$-adic shift it generates, and the measurable Bratteli-Vershik dynamical
system that it defines.
