The advent of Large Language Models (LLMs) has significantly reshaped the landscape of machine translation (MT), particularly for low-resource languages and domains that lack sufficient parallel corpora, linguistic tools, and computational infrastructure. This survey presents a comprehensive overview of recent progress in leveraging LLMs for MT. We analyze techniques such as few-shot prompting, cross-lingual transfer, and parameter-efficient fine-tuning that enable effective adaptation to under-resourced settings. The paper also explores synthetic data generation strategies using LLMs, including back-translation and lexical augmentation. Additionally, we compare LLM-based translation with traditional encoder-decoder models across diverse language pairs, highlighting the strengths and limitations of each. We discuss persistent challenges such as hallucinations, evaluation inconsistencies, and inherited biases while also evaluating emerging LLM-driven metrics for translation quality. This survey offers practical insights and outlines future directions for building robust, inclusive, and scalable MT systems in the era of large-scale generative models.