Brain dynamics associated with design creativity tasks are largely unexplored. Despite significant strides, there is a limited understanding of the brain-behavior during design creation tasks. The objective of this paper is to review the concepts of creativity and design creativity as well as their differences, and to explore the brain dynamics associated with design creativity tasks using electroencephalography (EEG) as a neuroimaging tool. The paper aims to provide essential insights for future researchers in the field of design creativity neurocognition. It seeks to examine fundamental studies, present key findings, and initiate a discussion on associated brain dynamics. The review employs thematic analysis and a forward and backward snowball search methodology with specific inclusion and exclusion criteria to select relevant studies. This search strategy ensured a comprehensive review focused on EEG-based creativity and design creativity experiments. Different components of those experiments such as participants, psychometrics, experiment design, and creativity tasks, are reviewed and then discussed. The review identifies that while some studies have converged on specific findings regarding EEG alpha band activity in creativity experiments, there remain inconsistencies in the literature. The paper underscores the need for further research to unravel the interplays between these cognitive processes. This comprehensive review serves as a valuable resource for readers seeking an understanding of current literature, principal discoveries, and areas where knowledge remains incomplete. It highlights both positive and foundational aspects, identifies gaps, and poses lingering questions to guide future research endeavors.