Crystal structures of two metal-organic frameworks (MFU-1 and MFU-2) are presented, both of which contain redox-active Co(II) centres coordinated by linear 1,4-bis[(3,5-dimethyl)pyrazol-4-yl] ligands. In contrast to many MOFs reported previously, these compounds show excellent stability against hydrolytic decomposition. Catalytic turnover is achieved in oxidation reactions by employing tert-butyl hydroperoxide and the solid catalysts are easily recovered from the reaction mixture. Whereas heterogeneous catalysis is unambiguously demonstrated for MFU-1, MFU-2 shows catalytic activity due to slow metal leaching, emphasising the need for a deeper understanding of structure-reactivity relationships in the future design of redox-active metal-organic frameworks. Mechanistic details for oxidation reactions employing tert-butyl hydroperoxide are studied by UV/Vis and IR spectroscopy and XRPD measurements. The catalytic process accompanying changes of redox states and structural changes were investigated by means of cobalt K-edge X-ray absorption spectroscopy. To probe the putative binding modes of molecular oxygen, the isosteric heats of adsorption of O(2) were determined and compared with models from DFT calculations. The stabilities of the frameworks in an oxygen atmosphere as a reactive gas were examined by temperature-programmed oxidation (TPO). Solution impregnation of MFU-1 with a co-catalyst (N-hydroxyphthalimide) led to NHPI@MFU-1, which oxidised a range of organic substrates under ambient conditions by employing molecular oxygen from air. The catalytic reaction involved a biomimetic reaction cascade based on free radicals. The concept of an entatic state of the cobalt centres is proposed and its relevance for sustained catalytic activity is briefly discussed.