Test-time adaptation (TTA) aims to adapt the neural network to the distribution of the target domain using only unlabeled test data. Most previous TTA methods have achieved success under mild conditions, such as considering only a single or multiple independent static domains. However, in real-world settings, the test data is sampled in a correlated manner and the test environments undergo continual changes over time, which may cause previous TTA methods to fail in practical noise scenarios, i.e., independent noise distribution shifts, continual noise distribution shifts, and continual mixed distribution shifts. To address these issues, we elaborate a Stable Test-time Adaptation Framework, called STAF, to stabilize the adaptation process. Specifically, to boost model robustness to noise distribution shifts, we present a multi-stream perturbation consistency method, enabling weak-to-strong views to be consistent, guided by the weak view from the original sample. Meanwhile, we develop a reliable memory-based corrector which utilizes reliable snapshots between the anchor model and the adapt model to correct prediction bias. Furthermore, we propose a dynamic parameter restoration strategy to alleviate error accumulation and catastrophic forgetting that takes into account both the distribution shift and sample adaptation degree. Extensive experiments demonstrate the robustness and effectiveness of STAF, which pushes the boundaries of test-time adaptation to more realistic scenarios and paves the way for stable deployment of real-world applications.