Deep neural networks achieve remarkable performance when training and test data share the same distribution, but this assumption often fails in real-world scenarios where data experiences continual distributional shifts. Continual Test-Time Adaptation (CTTA) tackles this challenge by adapting pretrained models to non-stationary target distributions on-the-fly, without access to source data or labeled targets, while addressing two critical failure modes: catastrophic forgetting of source knowledge and error accumulation from noisy pseudo-labels over time. In this comprehensive survey, we formally define the CTTA problem, analyze the diverse continual domain shift patterns that arise under different evaluation protocols, and propose a hierarchical taxonomy categorizing existing methods into three families: optimization-based strategies (entropy minimization, pseudo-labeling, parameter restoration), parameter-efficient methods (normalization layer adaptation, adaptive parameter selection), and architecture-based approaches (teacher-student frameworks, adapters, visual prompting, masked modeling). We systematically review representative methods in each category and provide comparative benchmarks and experimental results across standard evaluation settings. Finally, we discuss the limitations of current approaches and highlight emerging research directions, including adaptation of foundation models and black-box systems, offering a roadmap for future work in robust continual test-time adaptation, with further resources available at https://github.com/sarthaxxxxx/Awesome-Continual-Test-Time-Adaptation.