Reasoning has become one of the central capabilities driving modern large language models, yet turning that capability into reliable, efficient, and reproducible practice is far from solved. Further, increasingly, the bottlenecks are as much about data and systems as about models themselves. This lecture traces the landscape of LLM reasoning from foundations to frontiers. We begin with what "reasoning" means for an LLM and how chain-of-thought prompting reshaped the field by externalising intermediate steps. We then examine test-time scaling, the family of techniques (self-consistency, tree and graph search, reasoning-tuned models, etc.) that trade inference compute for answer quality. As a concrete case study, I will present Fleet of Agents (FoA), where swarms of lightweight agents explore search spaces in parallel and are resampled through a genetic-style process, showing that orchestration often matters more than sheer model size. Next, we turn to evaluation. Single-run accuracy hides as much as it reveals: identical prompts yield divergent answers, and headline numbers mask large run-to-run swings in both quality and cost. I will introduce ReasonBench, a variance-aware benchmark that treats reproducibility as a first-class concern and exposes the hidden instability of popular reasoning strategies. We close with efficiency, where data management meets Green AI. As reasoning workloads scale, the financial and environmental cost of redundant inference is becoming impossible to ignore. I will present CacheSaver, a modular client-side framework whose namespace-aware caching mechanism reduces cost and carbon emissions while preserving the statistical integrity of experiments — making large-scale reasoning research both more affordable and more sustainable. By the end of the lecture, attendees will have a clear conceptual map of how to select inference-time reasoning strategies for a given task, how to evaluate them with statistical rigour, and how to reason about their efficiency and deployment trade-offs, as well as a sense of where the open problems sit at the intersection of language models and data management.