Despite sequences being core to NLP, scant work has considered how to handle noisy sequence labels from multiple annotators for the same text. Given such annotations, we consider two complementary tasks: (1) aggregating sequential crowd labels to infer a best single set of consensus annotations; and (2) using crowd annotations as training data for a model that can predict sequences in unannotated text. For aggregation, we propose a novel Hidden Markov Model variant. To predict sequences in unannotated text, we propose a neural approach using Long Short Term Memory. We evaluate a suite of methods across two different applications and text genres: Named-Entity Recognition in news articles and Information Extraction from biomedical abstracts. Results show improvement over strong baselines. Our source code and data are available online.