Reasoning has become a defining capability of modern foundation models, yet its development in the audio modality remains limited. Audio poses challenges that are distinct from those of text and vision. It is continuous, temporally dense, and contains linguistic, paralinguistic, and environmental information at multiple time scales. As a result, audio reasoning models must align acoustic signals with the discrete semantic space of large language models, while still preserving fine-grained information needed for reliable inference. Progress is also limited by three major obstacles: the scarcity of genuinely audio-grounded reasoning data, shortcut learning and modality hallucination, and the tension between reasoning depth and real-time latency in spoken interaction. In this paper, we present the first dedicated survey of audio reasoning. We provide a unified formulation that distinguishes direct predictive modeling from reasoning-augmented generation, review the architectural and training foundations of audio reasoning models, and systematically organize recent advances in Audio-to-Text, Audio-to-Speech, Audio-Visual Reasoning and Agentic Audio Reasoning. We further examine emerging paradigms such as Chain-of-Thought prompting, supervised fine-tuning, reinforcement learning, and latency-aware spoken interaction, and discuss evaluation practices, open challenges, and future directions. Our goal is to offer a coherent roadmap for developing robust, efficient, and natively grounded audio reasoning systems.