This paper introduces Memory-limited Online Subspace Estimation Scheme (MOSES) for both estimating the principal components of data and reducing its dimension. More specifically, consider a scenario where the data vectors are presented sequentially to a user who has limited storage and processing time available, for example in the context of sensor networks. In this scenario, MOSES maintains an estimate of leading principal components of the data that has arrived so far and also reduces its dimension. In terms of its origins, MOSES slightly generalises the popular incremental Singular Value Decomposition (SVD) to handle thin blocks of data. This simple generalisation is in part what allows us to complement MOSES with a comprehensive statistical analysis that is not available for incremental SVD, despite its empirical success. This generalisationalso enables us to concretely interpret MOSES as an approximate solver for the underlying non-convex optimisation program. We also find that MOSES shows state-of-the-art performance in our numerical experiments with both synthetic and real-world datasets.