We review the structural chemistry and reactivity of copper-exchanged molecular sieves with chabazite (CHA) topology, as an industrially applied catalyst in ammonia mediated reduction of harmful nitrogen oxides (NH3-SCR) and as a general model system for red-ox active materials (also the recent results in the direct conversion of methane to methanol are considered). Notwithstanding the apparent structural simplicity of the material, a crystalline zeolite with only one crystallographically independent T site, the Cu-SSZ-13 catalyst reveals a high degree of complexity that has been decrypted by state of the art characterization tools. From the reviewed data, the following important aspects in the understanding of the Cu-SSZ-13 catalyst clearly emerged: (i) the structural dynamics of the Cu-species require precise control of the environmental conditions during activation and characterization; (ii) the availability of a large library of well-defined catalysts with different Si/Al and Cu/Al compositional ratios is key in unravelling the red-ox properties of the active Cu sites; (iii) a multi-technique approach is required, combining complementary techniques able to provide independent structural, electronic and vibrational information; (iv) synchrotron radiation based techniques (EXAFS, XANES, XES and time-resolved powder XRD) played a relevant role; (v) operando methodology (possibly supported by advanced chemometric approaches) is essential in obtaining structure–reactivity relations; (vi) the support of theoretical studies has been indispensable for the interpretation of the experimental output from characterization and for a critical assessment of mechanistic models. The old literature that classified Cu-exchanged zeolites in the category of single-site catalysts has been partially disproved by the recent advanced studies where it has been shown that the active site in the low temperature NH3-SCR catalyst is a mobile Cu-molecular entity that “lives in symbiosis” with an inorganic solid framework. Only in the high temperature NH3-SCR regime do the mobile Cu-species lose their ligands and find docking sites at the internal walls of the zeolite framework, thus reflecting the idea of a single-site catalyst. After a brief introduction, the review is divided into three main parts devoted to characterization (Section 2), reactivity (Section 3), and industrial applications (Section 4), followed by some concluding remarks and providing a perspective of the field.