In real-world applications, deep learning models often run in non-stationary environments where the target data distribution continually shifts over time. There have been numerous domain adaptation (DA) methods in both online and offline modes to improve cross-domain adaptation ability. However, these DA methods typically only provide good performance after a long period of adaptation and perform poorly on new domains before and during adaptation, especially when domain shifts happen suddenly and momentarily. On the other hand, domain generalization (DG) methods have been proposed to improve the model generalization ability on unadapted domains. However, existing DG works are ineffective for continually changing domains due to severe catastrophic forgetting of learned knowledge. To overcome these limitations of DA or DG in tackling continual domain shifts, we propose Twofer, a framework that simultaneously achieves target domain generalization (TDG), target domain adaptation (TDA), and forgetting alleviation (FA). Twofer includes a training-free data augmentation module to prepare data for TDG, a novel pseudo-labeling mechanism to provide reliable supervision for TDA, and a prototype contrastive alignment algorithm to align different domains for achieving TDG, TDA, and FA. Extensive experiments on Digits, PACS, and Domain Net datasets demonstrate that Twofer substantially outperforms state-of-the-art works in Continual DA, Source-Free DA, Test-Time/Online DA, Single DG, Multiple DG, and Unified DA DG. We envision this work as a significant milestone in tackling continual data domain shifts, with improved performance across target domain generalization, adaptation, and forgetting alleviation abilities.