Electroencephalography (EEG) provides a non-invasive approach for capturing brain dynamics and has become a cornerstone in clinical diagnostics, cognitive neuroscience, and neuroengineering. The inherent complexity, low signal-to-noise ratio, and variability of EEG signals have historically posed substantial challenges for interpretation. In recent years, artificial intelligence (AI), encompassing both classical machine learning (ML) and advanced deep learning (DL) methodologies, has transformed EEG analysis by enabling automatic feature extraction, robust classification, regression-based state estimation, and synthetic data generation. This survey synthesizes developments up to 2025, structured along three dimensions. The first dimension is task category, e.g., classification, regression, generation and augmentation, clustering and anomaly detection. The second dimension is the methodological framework, e.g., shallow learners, Convolutional Neural Networks (CNNs), Recurrent Neural Networks (RNNs), Transformers, Graph Neural Networks (GNNs), and hybrid approaches. The third dimension is application domain, e.g., neurological disease diagnosis, brain-computer interfaces (BCIs), affective computing, cognitive workload monitoring, and specialized tasks such as sleep staging and artifact removal. Publicly available EEG datasets and benchmarking initiatives that have catalyzed progress were reviewed in this study. The strengths and limitations of current AI models were critically evaluated, including constraints related to data scarcity, inter-subject variability, noise sensitivity, limited interpretability, and challenges of real-world deployment. Future research directions were highlighted, including federated learning (FL) and privacy-preserving learning, self-supervised pretraining of Transformer-based architectures, explainable artificial intelligence (XAI) tailored to neurophysiological signals, multimodal fusion with complementary biosignals, and the integration of lightweight on-device AI for continuous monitoring. By bridging historical foundations with cutting-edge innovations, this survey aims to provide a comprehensive reference for advancing the development of accurate, robust, and transparent AI-driven EEG systems.