Evidence for the hypothesis that input to the apical tufts of neocortical pyramidal cells plays a central role in cognition by amplifying their responses to feedforward input is reviewed. Apical tufts are electrically remote from the soma, and their inputs come from diverse sources including direct feedback from higher cortical regions, indirect feedback via the thalamus, and long-range lateral connections both within and between cortical regions. This suggests that input to tuft dendrites may amplify the cell’s response to basal inputs that they receive via layer 4 and which have synapses closer to the soma. ERP data supporting this inference is noted. Intracellular studies of apical amplification (AA) and of disamplification by inhibitory interneurons targeted only at tufts are reviewed. Cognitive processes that have been related to them by computational, electrophysiological, and psychopathological studies are then outlined. These processes include: figure-ground segregation and Gestalt grouping; contextual disambiguation in perception and sentence comprehension; priming; winner-take-all competition; attention and working memory; setting the level of consciousness; cognitive control; and learning. It is argued that theories in cognitive neuroscience should not assume that all neurons function as integrate-and-fire point processors, but should use the capabilities of cells with distinct sites of integration for driving and modulatory inputs. Potentially ‘unifying’ theories that depend upon these capabilities are reviewed. It is concluded that evolution of the primitives of AA and disamplification in neocortex may have extended cognitive capabilities beyond those built from the long-established primitives of excitation, inhibition, and disinhibition.