Our most distinguishing higher cognitive functions are controlled by the cerebral cortex. Comparative studies detail abundant anatomical and cellular features unique to the human developing and adult neocortex. Emerging genomic studies have further defined vast differences distinguishing developing human neocortices from related primates. These human-specific changes can affect gene function and/or expression, and result from structural variations such as chromosomal deletions and duplications, or from point mutations in coding and noncoding regulatory regions. Here, we review this rapidly growing field which aims to identify and characterize genetic loci unique to the human cerebral cortex. We catalog known human-specific genomic changes distinct from other primates, including those whose function has been interrogated in animal models. We also discuss how new model systems and technologies such as single cell RNA sequencing, primate iPSCs, and gene editing, are enabling the field to gain unprecedented resolution into function of these human-specific changes. Some neurological disorders are thought to uniquely present in humans, thus reinforcing the need to comprehensively understand human-specific gene expression in the developing brain.