  The spaces $A^s_{p,q}({\mathbb R}^n)$ with $A \in \{B,F \}$, $s\in {\mathbb
R}$ and $0<p,q \le \infty$ are usually introduced in terms of
Fourier--analytical decompositions. Related characterizations based on atoms
and wavelets are known nowadays in a rather final way. Quarks atomize the atoms
into constructive building blocks. It is the main aim of these notes to raise
quarkonial decompositions to the same level as related representations of the
spaces $A^s_{p,q}({\mathbb R}^n)$ in terms of atoms or wavelets. This will be
complemented by some applications. In addition we deal also with quarks in
domains and their relations to so--called refined localization spaces.
