This paper is devoted to developing the necessary orbital theory for predicting the lifetimes of satellites moving in an oblate atmosphere and subjected to the perturbations due to odd zonal harmonics in the geopotential. The effects of odd zonal harmonics and atmospheric oblateness are expressed as multiplying factors, F(oz) and F(ao), to be applied to the lifetime predictions calculated in the absence of the perturbations. The results are valid for the great majority of orbits about the Earth, and in particular for all orbital eccentricities between 0 and 1; but the limits set for the controlling parameters exclude (a) near-polar orbits with perigee heights lower than about 180 km, and (b) orbits having inclinations within a narrow band centred on 63.4$^\circ$. The results show that, when the controlling parameters are at their upper limits, either F(oz) or F(ao) can change the lifetime by up to about 35%, and taken together they can produce changes of up to 60%, if the initial and final positions of perigee are at specified points on the orbit and the eccentricity exceeds 0.2. Such combinations of values rarely arise, however, and the effects are more often of order 10-20%. Even at these moderate levels, the effects need to be taken into account in order to make realistic estimates of the decay dates of satellites in the last few months of their lives.