As Large Language Models (LLMs) push the boundaries of AI capabilities, their demand for data is growing. Much of this data is private and distributed across edge devices, making Federated Learning (FL) a de-facto alternative for fine-tuning (i.e., FedLLM). However, it faces significant challenges due to the inherent heterogeneity among clients, including varying data distributions and diverse task types. Towards a versatile FedLLM, we replace traditional dense model with a sparsely-activated Mixture-of-Experts (MoE) architecture, whose parallel feed-forward networks enable greater flexibility. To make it more practical in resource-constrained environments, we present FedMoE, the efficient personalized FL framework to address data heterogeneity, constructing an optimal sub-MoE for each client and bringing the knowledge back to global MoE. FedMoE is composed of two fine-tuning stages. In the first stage, FedMoE simplifies the problem by conducting a heuristic search based on observed activation patterns, which identifies a suboptimal submodel for each client. In the second stage, these submodels are distributed to clients for further training and returned for server aggregating through a novel modular aggregation strategy. Meanwhile, FedMoE progressively adjusts the submodels to optimal through global expert recommendation. Experimental results demonstrate the superiority of our method over previous personalized FL methods.