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We consider robust principal components analysis (PCA) based on multivariate MM estimators. We first study the robustness and efficiency of these estimators, particularly in terms of eigenvalues and eigenvectors. We then focus on inference procedures based on a fast and robust bootstrap for MM estimators. This method is an alternative to the approach based on the asymptotic distribution of the estimators and can also be used to assess the stability of the principal components. A formal consistency proof for the bootstrap method is given, and its finite-sample performance is investigated through simulations. We illustrate the use of the robust PCA and the bootstrap inference on a real dataset.