Brain organoids are proving to be physiologically relevant models for studying human brain development in terms of temporal transcriptional signature recapitulation, dynamic cytoarchitectural development, and functional electrophysiological maturation. Several studies have employed brain organoid technologies to elucidate human-specific processes of brain development, gene expression, and cellular maturation by comparing human-derived brain organoids to those of non-human primates (NHPs). Brain organoids have been established from a variety of NHP pluripotent stem cell (PSC) lines and many protocols are now available for generating brain organoids capable of reproducibly representing specific brain region identities. Innumerous combinations of brain region specific organoids derived from different human and NHP PSCs, with CRISPR-Cas9 gene editing techniques and strategies to promote advanced stages of maturation, will successfully establish complex brain model systems for the accurate representation and elucidation of human brain development. Identified human-specific processes of brain development are likely vulnerable to dysregulation and could result in the identification of therapeutic targets or disease prevention strategies. Here, we discuss the potential of brain organoids to successfully model human-specific processes of brain development and explore current strategies for pinpointing these differences.