Recent studies have disclosed several oscillations occurring during resting sleep within the frequency range of the classical delta band (0.5-4 Hz). There are at least 3 oscillations with distinct mechanisms and sites of origin: a slow (<1 Hz) cortically-generated oscillation, a clock-like thalamic oscillation (1-4 Hz), and a cortical oscillation (1-4 Hz). The present paper reviews data on these oscillations and the possible mechanisms which coalesce them into the polymorphic waves of slow wave sleep. Data stem from intracellular (over 500 single cell and 50 double impalements) and field potentials recorded from the cortex and thalamus of cats (120 animals) under ketamine and xylazine anesthesia. Other experiments were based on whole night EEG recordings from humans (5 subjects). The frequency of the slow oscillation both in anesthetized animals and in naturally sleeping humans ranged between 0.1 and 1 Hz (89% of the cases being between 0.5 and 0.9 Hz). The slow (<1 Hz) oscillation is reflected in the EEG as rhythmic sequences of surface-negative waves (associated with hyperpolarizations of deeply-lying neurons) and surface-positive K-complexes (representing excitation in large pools of cortical neurons). Through its long-range synchronization, the slow oscillation has the ability to trigger and to group thalamically-generated spindles and two delta (1-4 Hz) oscillations. Finally, it is argued that the analysis of the electroencephalogram should transcend the spectral analyses, by taking into account the shape of the waves and, when possible, the basic mechanisms that generate those waves.