Large Language Models (LLMs) have shown promising results across various tasks, yet their reasoning capabilities remain a fundamental challenge. Developing AI systems with strong reasoning capabilities is regarded as a crucial milestone in the pursuit of Artificial General Intelligence (AGI) and has garnered considerable attention from both academia and industry. Various techniques have been explored to enhance the reasoning capabilities of LLMs, with neuro-symbolic approaches being a particularly promising way. This paper comprehensively reviews recent developments in neuro-symbolic approaches for enhancing LLM reasoning. We first present a formalization of reasoning tasks and give a brief introduction to the neuro-symbolic learning paradigm. Then, we discuss neuro-symbolic methods for improving the reasoning capabilities of LLMs from three perspectives: Symbolic->LLM, LLM->Symbolic, and LLM+Symbolic. Finally, we discuss several key challenges and promising future directions. We have also released a GitHub repository including papers and resources related to this survey: https://github.com/LAMDASZ-ML/Awesome-LLM-Reasoning-with-NeSy.