Most approaches to out-of-distribution (OOD) generalization learn domain-invariant representations by discarding contextual bias. In this paper, we raise a critical question: Should bias be eliminated? If not, is there a general way to leverage bias for better OOD generalization? To answer these questions, we first provide a theoretical analysis that characterizes the circumstances in which biased features contribute positively. Although theoretical results show that bias may sometimes play a positive role, leveraging it effectively is non-trivial, since its harmful and beneficial components are often entangled. Recent advances have sought to refine the prediction of bias by presuming reliable predictions from invariant features. However, such assumptions may be too strong in the real world, especially when the target also shifts from training to testing domains. Motivated by this challenge, we introduce a framework to leverage bias in a more general scenario. Specifically, we employ a generative model to capture the data generation process and identify the underlying bias factors, which are then used to construct a bias-aware predictor. Since the bias-aware predictor may shift across environments, we first estimate the environment state to train predictors under different environments, combining them as a mixture of domain experts for the final prediction. Then, we build a general invariant predictor, which can be invariant under label shift to guide the adaptation of the bias-aware predictor. Evaluations on synthetic data and standard domain generalization benchmarks demonstrate that our method consistently outperforms both invariance only baselines, recent bias utilization approaches and advanced baselines, yielding improved robustness and adaptability.