This paper presents a comprehensive survey of the field of compositional visual reasoning (CVR), analyzing over 260 papers published from 2023 to 2025. CVR aims to empower machines to decompose visual scenes and perform multi-step logical reasoning based on intermediate concepts, much like humans. We define the advantages of compositional approaches (cognitive alignment, semantic fidelity, robustness, interpretability, and data efficiency) and trace five paradigm shifts: from prompt-based, language-centric pipelines to tool-based LLMs and VLMs, thought-chain reasoning, and integrated agent VLMs. We present over 60 benchmarks and metrics, highlighting key insights, challenges (e.g., limitations of LLM-based reasoning, hallucinations, biases in deductive reasoning, scalable supervision, tool integration, and benchmark limitations), and future directions (e.g., world model integration, human-AI collaborative reasoning, and richer evaluation protocols).