Many machine learning systems are vulnerable to small perturbations made to inputs either at test time or at training time. This has received much recent interest on the empirical front due to applications where reliability and security are critical. However, theoretical understanding of algorithms that are robust to adversarial perturbations is limited. In this work we focus on Principal Component Analysis (PCA), a ubiquitous algorithmic primitive in machine learning. We formulate a natural robust variant of PCA where the goal is to find a low dimensional subspace to represent the given data with minimum projection error, that is in addition robust to small perturbations measured in $\ell_q$ norm (say $q=\infty$). Unlike PCA which is solvable in polynomial time, our formulation is computationally intractable to optimize as it captures a variant of the well-studied sparse PCA objective as a special case. We show the following results: -Polynomial time algorithm that is constant factor competitive in the worst-case with respect to the best subspace, in terms of the projection error and the robustness criterion. -We show that our algorithmic techniques can also be made robust to adversarial training-time perturbations, in addition to yielding representations that are robust to adversarial perturbations at test time. Specifically, we design algorithms for a strong notion of training-time perturbations, where every point is adversarially perturbed up to a specified amount. -We illustrate the broad applicability of our algorithmic techniques in addressing robustness to adversarial perturbations, both at training time and test time. In particular, our adversarially robust PCA primitive leads to computationally efficient and robust algorithms for both unsupervised and supervised learning problems such as clustering and learning adversarially robust classifiers.