The problem of segmenting linearly ordered data is frequently encountered in time-series analysis, computational biology, and natural language processing. Segmentations obtained independently from replicate data sets or from the same data with different methods or parameter settings pose the problem of computing an aggregate or consensus segmentation. This Segmentation Aggregation problem amounts to finding a segmentation that minimizes the sum of distances to the input segmentations. It is again a segmentation problem and can be solved by dynamic programming. The aim of this contribution is (1) to gain a better mathematical understanding of the Segmentation Aggregation problem and its solutions and (2) to demonstrate that consensus segmentations have useful applications. Extending previously known results we show that for a large class of distance functions only breakpoints present in at least one input segmentation appear in the consensus segmentation. Furthermore, we derive a bound on the size of consensus segments. As show-case applications, we investigate a yeast transcriptome and show that consensus segments provide a robust means of identifying transcriptomic units. This approach is particularly suited for dense transcriptomes with polycistronic transcripts, operons, or a lack of separation between transcripts. As a second application, we demonstrate that consensus segmentations can be used to robustly identify growth regimes from sets of replicate growth curves.