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**Graph Neural Networks (GNN)** is **a class of neural network architectures designed to process graph-structured data through message passing between nodes — enabling learning on irregular structures and graph-level predictions while naturally handling variable-size inputs**. Graph Neural Networks extend deep learning to non-Euclidean domains where data naturally form graphs or networks. The core principle of GNNs is message passing: each node iteratively updates its representation by aggregating information from its neighbors. In a typical GNN layer, each node computes messages based on its own features and neighbors' features, aggregates these messages (typically via summation, mean, or max operation), and passes the aggregated information through a neural network to produce updated node representations. This formulation naturally handles graphs with variable numbers of nodes and edges. Different GNN architectures make different choices about how to compute and aggregate messages. Graph Convolutional Networks (GCN) aggregate features through a spectral filter approximation, operating efficiently in vertex space. Graph Attention Networks (GAT) learn attention weights over neighbors, enabling selective message passing based on relevance. GraphSAGE samples a fixed-size neighborhood and aggregates features, enabling scalability to very large graphs. Message Passing Neural Networks (MPNN) provide a unified framework encompassing these variants. Spectral approaches operate on the graph Laplacian eigenvalues, connecting to classical harmonic analysis on graphs. GNNs naturally express permutation invariance — their predictions don't depend on node ordering — and handle irregular structures that convolutional and recurrent approaches struggle with. Applications span molecular property prediction, social network analysis, recommendation systems, and knowledge graph reasoning. Node-level tasks predict node labels, edge-level tasks predict edge properties, and graph-level tasks produce single outputs for entire graphs. Graph pooling operations progressively coarsen graphs while preserving relevant structural information. GNNs have proven effective for out-of-distribution generalization, sometimes outperforming fully connected networks trained on explicit feature representations. Limitations include shallow architectures (many GNN layers hurt performance due to over-squashing), lack of theoretical understanding of expressiveness, and challenges with very large graphs. Recent work addresses these through deeper GNNs, theoretical analysis via Weisfeiler-Lehman tests, and sampling-based scalability approaches. **Graph Neural Networks enable deep learning on non-Euclidean structured data, with message passing providing an elegant framework for learning representations on graphs and networks.**