The electronic and geometrical structures of the ground and some excited states of the FeOn and FeOn- clusters (n = 1-4) have been calculated using the density-functional theory with generalized gradient approximation for the exchange-correlation potential. It is found that the multiplicity of the ground states decreases with increasing n, and the ground states of FeO- and FeO2- are quartets whereas those of FeO3- and FeO4-are doublets. All of these anions possess isomers with different spatial or spin symmetries that are close in energy to their ground states. For example, FeO4- has at least five stationary states that are stable against electron detachment and fragmentation. Our calculated adiabatic electron affinities (A(ad)) Of FeO, FeO2, and FeO3 are within 0.2 eV of the experiment. FeO4 was found to be a particularly interesting cluster. Although its neutral precursor possesses a closed electronic shell structure, it has an A(ad) of 3.8 eV, which is higher than the electronic affinity of halogen atoms. The experimental estimate of 3.3 eV for the A(ad) of FeO4 is shown to originate from the detachment of an electron from one of the higher-energy isomers of the FeO4- cluster. The energetically preferred dissociation channels of FeO2, FeO3, FeO4, FeO3-, and FeO4- correspond to abstraction of an O-2 dimer but not to an Fe-O bond rupture. FeO3- and FeO4- are found to be thermodynamically more stable than their neutral closed-shell parents, and FeO3- is the most stable of all the clusters studied. The existence of several low-lying states with different multiplicities in FeOn and FeOn- indicates that their magnetic properties may strongly depend on temperature.