The topography of the EEG of human neonates is studied in terms of its power spectral density and its estimated complexity as a function of both the postmenstrual age (PMA) and the sleep state. The monopolar EEGs of three groups of seven neonates (preterm, term and older term) were recorded during active (AS) and quiet sleep (QS) from electrodes Fp1, Fp2, T3, T4, C3, C4, O1 and O2. The existence of changes between groups and sleep states in the power of delta, theta, alpha and beta bands and in the dimensional complexity of these electrodes was tested. Additionally, the nonlinearity of the EEG in each electrode and situation was analyzed. The results of the spectral measures show an increment of the power in the low frequency bands from AS to QS and with the PMA, which can be mainly traced on central and temporal electrodes. This change is shown as well by the dimensional complexity, which also presents the greatest differences in the central derivations. Moreover, the signals show evidence of nonlinearity in almost all the groups and situations, although a dynamic change from nonlinear to linear character is apparent in the central electrodes with increased PMA. As a result, it is concluded that nonlinear analysis methods provide a clear portrait of the integrated brain activity that complements the information of spectral analysis in the characterization of the brain development and the sleep states in neonates.