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In recent years, supervised learning-based methods have achieved excellent performance for hyperspectral image (HSI) classification. However, the collection of training samples with labels is not only costly but also time-consuming. This fact usually causes the existence of weak supervision, including incorrect supervision where mislabeled samples exist and incomplete supervision where unlabeled samples exist. Focusing on the inaccurate supervision and incomplete supervision, the weakly supervised classification of HSI is investigated in this paper. For inaccurate supervision, complementary learning (CL) is firstly introduced for HSI classification. Then, a new method, which is based on selective CL and convolutional neural network (SeCL-CNN), is proposed for classification with noisy labels. For incomplete supervision, a data augmentation-based method, which combines mixup and Pseudo-Label (Mix-PL) is proposed. And then, a classification method, which combines Mix-PL and CL (Mix-PL-CL), is designed aiming at better semi-supervised classification capacity of HSI. The proposed weakly supervised methods are evaluated on three widely-used hyperspectral datasets (i.e., Indian Pines, Houston, and Salinas datasets). The obtained results reveal that the proposed methods provide competitive results compared to the state-of-the-art methods. For inaccurate supervision, the proposed SeCL-CNN has outperformed the state-of-the-art method (i.e., SSDP-CNN) by 0.92%, 1.84%, and 1.75% in terms of OA on the three datasets, when the noise ratio is 30%. And for incomplete supervision, the proposed Mix-PL-CL has outperformed the state-of-the-art method (i.e., AROC-DP) by 1.03%, 0.70%, and 0.82% in terms of OA on the three datasets, with 25 training samples per class. [ABSTRACT FROM AUTHOR]