Data fusion and multi-cue data matching are fundamental tasks arising in a variety of systems that process large amounts of data. These tasks often rely on dimensionality reduction techniques that traditionally follow a data acquisition/reprocessing phase. In this talk, I will describe a powerful framework based on diffusions that can be used in order to learn the intrinsic geometry of data sets. These techniques allow to simultaneously handle data acquisition issues and data processing tasks. In particular, I will explain how we can use this set of tools in order to address three major challenges related to data fusion: 1) How to deal with data coming from sensors/sources sampled at different rates, and possibly at different times. We provide algorithms to obtain density-invariant descriptors (parametrization) of data sets. 2) How to integrate and combine information streams coming from different sensors into one representation of the data. The diffusion coordinates allow to learn the geometry of the data captured by each sensor independently, and then to combine the various representations into a unified description of the data. 3) How to do matching of data sets based on their intrinsic geometry. As an illustration, I will present numerical results on the integration of audio and video streams for lip-reading and speech recognition. Other examples will be more focused on imaging (multiscale data-driven image segmentation, image data sets alignment). This is joint work with R.R. Coifman, A. Glaser, Y. Keller and S.W. Zucker (Yale university).