  We introduce the notion of self-similarity for compact quantum groups. For a
finite set $X$, we introduce a $C^*$-algebra $\mathbb{A}_X$, which is the
quantum automorphism group of the infinite homogeneous rooted tree $X^*$.
Self-similar quantum groups are then certain quantum subgroups of
$\mathbb{A}_X$. Our main class of examples are called finitely-constrained
self-similar quantum groups, and we find a class of these examples that can be
described as quantum wreath products by subgroups of the quantum permutation
group.
