A method for principal component analysis is proposed that is sparse and robust at the same time. The sparsity delivers principal components that have loadings on a small number of variables, making them easier to interpret. The robustness makes the analysis resistant to outlying observations. The principal components correspond to directions that maximize a robust measure of the variance, with an additional penalty term to take sparseness into account. We propose an algorithm to compute the sparse and robust principal components. The algorithm computes the components sequentially, and thus it can handle datasets with more variables than observations. The method is applied on several real data examples, and diagnostic plots for detecting outliers and for selecting the degree of sparsity are provided. A simulation experiment studies the effect on statistical efficiency by requiring both robustness and sparsity. Supplementary materials are available online on the journal web site.