Reasoning has become central to how Large Language Models (LLMs) are evaluated and interpreted, spanning Chain-of-Thought (CoT), mathematical problem-solving, multi-hop question answering, code generation, retrieval-augmented reasoning, tool use, and multimodal decision-making. In this survey, we introduce the Periodic Table of LLM Reasoning, a framework organizing 300+ recent papers by reasoning paradigm, methodological mechanism, evaluation setting, and failure mode. We classify LLM reasoning into nine paradigms: Chain-of-Thought, Multi-Hop, Mathematical, Commonsense, Visual and Temporal, Code and Algorithmic, Retrieval-Augmented, Tool-Augmented or Agentic, and Reinforcement Learning-based reasoning. For each, we review approaches, including prompting, architectural interventions, supervised fine-tuning, verifier-guided inference, reward modeling, retrieval, tool interfaces, agentic workflows, and benchmark design. We argue that LLM reasoning is not a single emergent capability but a family of scaffolded behaviors shaped by model scale, task structure, external memory, supervision, and evaluation protocols. We synthesize recurring failure modes, including hallucinated reasoning, brittle multi-step inference, spurious rationales, weak causal grounding, poor out-of-distribution generalization, benchmark contamination, and unreliable self-verification. Progress is difficult to compare across paradigms because gains may arise from prompting, retrieval, verifier design, or benchmark structure rather than general reasoning ability. The survey connects methods to their assumptions, strengths, and failure modes, providing a reference map of the field and a diagnostic framework for future work. We conclude that robust LLM reasoning will require meta-reasoning, multimodal and temporal grounding, adaptive tool use, and principled evaluation under distribution shift.

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