Accent conversion has rapidly progressed alongside growing interest in improving global cross-cultural communication. This survey presents an overview of the evolution of accent conversion methodologies, analyzing how the field has developed in response to fundamental challenges related to data alignment, representation disentanglement, and resource scarcity. We trace the progression from early rule-based digital signal processing approaches such as spectral manipulation and formant-based analysis to modern neural architectures capable of flexible and reference-free accent transformation. In addition, the survey situates accent conversion within its linguistic foundations and examines how different application requirements impose varying constraints on the balance between accent modification and speaker identity preservation. Finally, it reviews commonly used speech datasets and evaluation methodologies, identifies persistent challenges, and outlines directions for future research aimed at achieving more controllable and perceptually consistent accent conversion.