<p>Electronic circuit design is also becoming more difficult as contemporary electronic systems are expected to be more demanding in terms of performance, reduced power consumption, higher integration density, reduced design cycles, and enhanced reliability. In this respect, generative Artificial Intelligence (AI) has become a promising suite of methods to automate design activities, explore large design spaces, enhance prediction accuracy, and optimize their work faster. Nonetheless, current literature is still divided, and the majority of studies concentrate on one of the types of models or a specific field of application, and it hard to have a single perspective regarding the application of generative AI in circuit design. In order to cover this gap, this paper provides a Systematic Literature Review (SLR) of generative AI applications in electronic circuit design, specifically Generative Adversarial Networks (GANs), Reinforcement Learning (RL), Variational Autoencoders (VAEs) and Graph Neural Networks (GNNs). We suggest a hierarchical taxonomy of these methods based on the main purpose in the design flow, such as design automation, performance optimization, anomaly detection, and intelligent modeling. Moreover, we also offer a comparative study of methods reviewed by analyzing strengths, weaknesses, conditions of application, and the possibility of complementing each other in terms of a complex design environment. The findings suggest that generative AI can make a significant contribution to various phases of circuit design, especially when there is a data constraint, and the problem is highly complex. This review provides a better structure of the study that researchers and practitioners can use to choose, analyze, and develop AI-based solutions to more efficient, scalable, and reliable electronic design.</p>