This paper presents an in-depth survey on the use of multimodal Generative Artificial Intelligence (GenAI) and autoregressive Large Language Models (LLMs) for human motion understanding and generation, offering insights into emerging methods, architectures, and their potential to advance realistic and versatile motion synthesis. Focusing exclusively on text and motion modalities, this research investigates how textual descriptions can guide the generation of complex, human-like motion sequences. The paper explores various generative approaches, including autoregressive models, diffusion models, Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), and transformer-based models, by analyzing their strengths and limitations in terms of motion quality, computational efficiency, and adaptability. It highlights recent advances in text-conditioned motion generation, where textual inputs are used to control and refine motion outputs with greater precision. The integration of LLMs further enhances these models by enabling semantic alignment between instructions and motion, improving coherence and contextual relevance. This systematic survey underscores the transformative potential of text-to-motion GenAI and LLM architectures in applications such as healthcare, humanoids, gaming, animation, and assistive technologies, while addressing ongoing challenges in generating efficient and realistic human motion.