In recent years, considerable research advancements have emerged in the application of inverse design methods to enhance the performance of electromagnetic metamaterials. Notably, the integration of deep learning (DL) technologies, with their robust capabilities in data analysis, categorization, and interpretation, has demonstrated revolutionary potential in optimization algorithms for improved efficiency. In this review, current inverse design methods for electromagnetic metamaterials are presented, including topology optimization, evolutionary algorithms, and DL-based methods. Their application scopes, advantages and limitations, as well as the latest research developments are respectively discussed. The classical iterative inverse design methods categorized topology optimization and evolutionary algorithms are discussed separately, for their fundamental role in solving inverse design problems. Also, attention is given on categories of DL-based inverse design methods, i.e. classifying into DL-assisted, direct DL, and physics-informed neural network methods. A variety of neural network architectures together accompanied by relevant application examples are highlighted, as well as the practical utility of these overviewed methods. Finally, this review provides perspectives on potential future research directions of electromagnetic metamaterials inverse design and integrated artificial intelligence methodologies.