A central problem in Interactive Question Answering (IQA) is how to answer Follow-Up Questions (FU Qs), possibly by taking advantage of information from the dialogue context. We assume that FU Qs can be classified into specific types which determine if and how the correct answer relates to the preceding dialogue. The main goal of this paper is to propose an empirically motivated typology of FU Qs, which we then apply in a practical IQA setting. We adopt a supervised machine learning framework that ranks answer candidates to FU Qs. Both the answer ranking and the classification of FU Qs is done in this framework, based on a host of measures that include shallow and deep inter-utterance relations, automatically collected dialogue management meta information, and human annotation. We use Principal Component Analysis (PCA) to integrate these measures. As a result, we confirm earlier findings about the benefit of distinguishing between topic shift and topic continuation FU Qs. We then present a typology of FU Qs that is more fine-grained, extracted from the PCA and based on real dialogue data. Since all our measures are automatically computable, our results are relevant for IQA systems dealing with naturally occurring FU Qs.