With the use of artificial intelligence (AI) for image analysis of Magnetic Resonance Imaging (MRI), the lack of training data has become an issue. Realistic synthetic MRI images can serve as a solution and generative models have been proposed. This study investigates the most recent advances on synthetic brain MRI image generation with AI-based generative models. A search has been conducted on the relevant studies published within the last three years, followed by a narrative review on the identified articles. Popular models from the search results have been discussed in this study, including Generative Adversarial Networks (GANs), diffusion models, Variational Autoencoders (VAEs), and transformers.