  The permutation group $S_N$ has a free analogue $S_N^+$, which is
non-classical and infinite at $N\geq4$. We review here the known basic facts on
$S_N^+$, with emphasis on algebraic and probabilistic aspects. We discuss as
well the structure of the closed subgroups $G\subset S_N^+$, with particular
attention to the quantum reflection groups.
