AdamW has become one of the most effective optimizers for training large-scale models. We have also observed its effectiveness in the context of federated learning (FL). However, directly applying AdamW in federated learning settings poses significant challenges: (1) due to data heterogeneity, AdamW often yields high variance in the second-moment estimate $\boldsymbol{v}$; (2) the local overfitting of AdamW may cause client drift; and (3) Reinitializing moment estimates ($\boldsymbol{v}$, $\boldsymbol{m}$) at each round slows down convergence. To address these challenges, we propose the first \underline{Fed}erated \underline{AdamW} algorithm, called \texttt{FedAdamW}, for training and fine-tuning various large models. \texttt{FedAdamW} aligns local updates with the global update using both a \textbf{local correction mechanism} and decoupled weight decay to mitigate local overfitting. \texttt{FedAdamW} efficiently aggregates the \texttt{mean} of the second-moment estimates to reduce their variance and reinitialize them. Theoretically, we prove that \texttt{FedAdamW} achieves a linear speedup convergence rate of $\mathcal{O}(\sqrt{(L Δσ_l^2)/(S K R ε^2)}+(L Δ)/R)$ without \textbf{heterogeneity assumption}, where $S$ is the number of participating clients per round, $K$ is the number of local iterations, and $R$ is the total number of communication rounds. We also employ PAC-Bayesian generalization analysis to explain the effectiveness of decoupled weight decay in local training. Empirically, we validate the effectiveness of \texttt{FedAdamW} on language and vision Transformer models. Compared to several baselines, \texttt{FedAdamW} significantly reduces communication rounds and improves test accuracy. The code is available in https://github.com/junkangLiu0/FedAdamW.