1 Changing Concepts of Mechanisms of Waking and Sleep States.- 2 Methodological Advances in Knowledge of Morphological Substrates Underlying States of Vigilance.- 3 The Origin of Afferents to the Brainstem Core.- 4 Efferent Connections of Brainstem Neurons.- 5 Intrinsic Electrophysiological Properties of Brainstem Neurons and Their Relationship to Behavioral Control.- 6 Neurotransmitter-Modulated Ionic Currents of Brainstem Neurons and Some of Their Targets.- 7 Synchronized Brain Oscillations and Their Disruption by Ascending Brainstem Reticular Influxes.- 8 Brainstem Ascending Systems Controlling Synaptic Transmission of Afferent Signals.- 9 Brainstem Genesis and Thalamic Transfer of Pontogeniculooccipital Waves.- 10 Motor The Brainstem Oculomotor System and Mechanisms of Motor Atonia in REM Sleep.- 11 Neuronal Control of the Sleep-Wake States.- 12 REM Sleep as a Biological Rhythm. The Phenomenology and a Structural and Mathematical Model.- 13 Brainstem Mechanisms of Dreaming and of Disorders of Sleep in Man.- References.