Artificial intelligence increasingly enters quantitative finance through option pric ing, volatility calibration, hedging, asset pricing, risk measurement, market simula tion, financial text analysis, and time-series forecasting. Yet finance differs from many prediction domains because a statistically accurate model can still be economically invalid. Prices must respect no-arbitrage restrictions, discounted asset prices must be consistent with martingale arguments under suitable measures, volatility surfaces must satisfy static and dynamic admissibility conditions, hedging rules must account for transaction costs and risk preferences, and scenario generators must preserve tail behavior rather than merely match average dependence. This semi-systematic review studies modern financial AI through this constraint-centered perspective. It organizes the literature from arbitrage pricing and stochastic volatility to neural option pricing, deep calibration, deep hedging, no-arbitrage asset pricing, generative market models, f inancial large language models, and time-series foundation models. The review makes three contributions. First, it separates financial constraints into structural admissibil ity, task-level economic objectives, and governance-validity requirements. Second, it identifies recurring design patterns for constraint injection, including hard parameter ization, soft penalties, projection or repair, adversarial testing, and temporal valida tion. Third, it proposes layered and task-specific evaluation criteria that distinguish statistical fit from mathematical admissibility, economic usefulness, robustness, and deployability. The main argument is that financial mathematics is not a legacy layer that deep learning replaces. It provides the inductive biases, loss functions, diagnostic tests, and deployment gates that make AI models usable in quantitative finance.