Significance: As onshore and nearshore resources become increasingly scarce, exploiting the deep sea has become a strategic priority for energy production, aquaculture, raw materials, and maritime transport. Deep-sea engineering relies on various offshore floating structures that operate under harsh, complex, and time-varying wind, wave, and current conditions. Accurate and efficient prediction of their motions and internal force responses is essential for structural safety, optimal design, and operational planning. Conventional methods, such as computational fluid dynamics and potential flow theory, are computationally expensive or imprecise when strong nonlinearities are present. Advances in sensors, computing power, and big data technology have enabled artificial intelligence (AI) applications in this field. Artificial neural networks (ANNs) adaptively capture complex nonlinear dynamics from large datasets, making AI-based response prediction an effective bridge between efficiency and accuracy in ocean engineering. This review surveys recent progress in AI methods for predicting the dynamic responses of offshore floating structures, underscoring their strengths and limitations and outlining future research directions. Progress: This review consolidates recent advances in applying AI to three key predictive tasks for offshore floating structures (e.g., oil and gas platforms, floating production, storage and offloading units (FPSOs), and floating wind turbines). First, in time-series prediction, recurrent ANNs, such as gated recurrent units and long short-term memory networks, are widely used for short-term forecasting of floater motions and mooring tensions. Current research primarily focuses on two key improvement strategies. The first involves optimizing input features, which include environmental time histories (e.g., wave elevation and wind speed) and dynamic response time series (e.g., floater motions and internal structural forces). The second focuses on integrating AI with complementary techniques. Signal processing algorithms, such as variational mode decomposition, are used to reduce the bandwidth of model inputs. Optimization algorithms, such as Bayesian optimization, are employed to fine-tune model hyperparameters. Furthermore, incorporating physical laws (e.g., hydrodynamic transfer functions) enhances the model's generalization capability. Second, for extreme-value prediction, ANNs such as multilayer perceptrons and backpropagation networks are trained to map environmental parameters directly to short-term extremes or to extreme-value distribution parameters, thereby greatly reducing computational cost compared with time-domain simulations. For long-term extremes, representative sea states are sampled, and a surrogate model is trained to rapidly predict short-term extremes; probability convolution across states then provides long-term estimates that approach the accuracy of traditional full long-term analyses at a fraction of the computational cost. Third, for short-term fatigue damage prediction, ANNs are applied in both frequency-domain analysis (e.g., approximating nonlinear stress transfer functions) and time-domain analysis (e.g., mapping environmental parameters directly to load ranges or damage equivalent loads). For long-term fatigue assessment, two practical strategies prevail. The first is similar to that used in long-term extreme predictions. The second employs active learning to iteratively select the most informative samples, considerably reducing the required number of simulations while preserving accuracy. Conclusions and Prospects: AI provides significant advantages in rapid prediction and effective modeling of strong nonlinearities, overcoming the limitations of traditional numerical methods and enabling efficient forecasting and design optimization. However, most models are purely data-driven and thus assume limited generalizability to unseen conditions, lack physical interpretability, and often ignore built-in uncertainty quantification. Additionally, while various time-series prediction methods have been extensively compared, similar cross-evaluations are scarce for extreme-value and fatigue prediction approaches. To translate AI advances into reliable engineering practice, future work should prioritize physics-informed neural networks that embed fundamental hydrodynamics to improve generalization and trustworthiness, integrate uncertainty quantification frameworks such as Bayesian neural networks for reliability-based design, and develop more efficient strategies for long-term extreme-value and fatigue prediction. Finally, establishing high-quality shared datasets, standardized benchmarks, and validation protocols will be essential to migrate these techniques from research prototypes to routine engineering tools, powering digital twins and forecasting systems for offshore floating structures.