  The $S$-adic conjecture claims that there exists a condition $C$ such that a
sequence has a sub-linear complexity if and only if it is an $S$-adic sequence
satisfying Condition $C$ for some finite set $S$ of morphisms. We present an
overview of the factor complexity of $S$-adic sequences and we give some
examples that either illustrate some interesting properties or that are
counter-examples to what could be believed to be "a good Condition $C$".
