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Although there is a great success of applying deep learning on a wide variety of tasks, it heavily relies on a large amount of labeled training data, which could be hard to obtain in many real scenarios. To address this problem, unsupervised and semi-supervised learning emerge to take advantage of the plenty of cheap unlabeled data to improve the model generalization. In this dissertation, we claim that equivariant and invariance are two critical criteria to approach robust unsupervised and semi-supervised learning. The idea is as follows: the features of a robust model ought to be sufficiently informative and equivariant to transformations on the input data, and the classifiers should be resilient and invariant to small perturbations on the data manifold and model parameters. Specifically, features are learnt via auto-encoding the transformations on the input data, and models are regularized through minimizing the effects of perturbations on features or model parameters. Experiments on several benchmarks show the proposed methods outperform many state-of-the-art approaches on unsupervised and semi-supervised learning, proving importance of the equivariance and invariance rules for robust feature representation learning.