Volcanic degassing of planetary interiors has important implications for their corresponding atmospheres. The oxidation state of rocky interiors affects the volatile partitioning during mantle melting and subsequent volatile speciation near the surface. Here we show that the mantle redox state is central to the chemical composition of atmospheres while factors such as planetary mass, thermal state, and age mainly affect the degassing rate. We further demonstrate that mantle oxygen fugacity has an effect on atmospheric thickness and that volcanic degassing is most efficient for planets between 2 and 4 Earth masses. We show that outgassing of reduced systems is dominated by strongly reduced gases such as H2, with only smaller fractions of moderately reduced/oxidised gases (CO, H2O). Overall, a reducing scenario leads to a lower atmospheric pressure at the surface and to a larger atmospheric thickness compared to an oxidised system. Atmosphere predictions based on interior redox scenarios can be compared to observations of atmospheres of rocky exoplanets, potentially broadening our knowledge on the diversity of exoplanetary redox states.