In recent years, with the development of deep learning, electroencephalogram (EEG) classification networks have achieved certain progress. Transformer-based models can perform well in capturing long-term dependencies in EEG signals. However, their quadratic computational complexity leads to significant computational overhead. Moreover, most EEG classification models are only suitable for single tasks, showing poor generalization capabilities across different tasks and further unable to handle EEG data from various tasks simultaneously due to variations in signal length and the number of channels. In this paper, we introduce a universal EEG classification network named EEGMamba, which seamlessly integrates the Spatio-Temporal-Adaptive (ST-Adaptive) module, Bidirectional Mamba, and Mixture of Experts (MoE) into a unified framework for multiple tasks. The proposed ST-Adaptive module performs unified feature extraction on EEG signals of different lengths and channel counts through spatio-adaptive convolution and incorporates a class token to achieve temporal-adaptability. Moreover, we design a bidirectional Mamba particularly suitable for EEG signals for further feature extraction, balancing high accuracy and fast inference speed in processing long EEG signals. In order to better process EEG data for different tasks, we introduce Task-aware MoE with a universal expert, achieving the capture of both differences and commonalities between EEG data from different tasks. We test our model on eight publicly available EEG datasets, and experimental results demonstrate its superior performance in four types of tasks: seizure detection, emotion recognition, sleep stage classification, and motor imagery. The code is set to be released soon.