Abstract:  Our understanding of the distant universe has been fundamentally limited by the difficulty and expense of imaging large numbers of high-redshift galaxies.  However, while most galaxies are too faint to be detected individually, we can detect their presence in the aggregate by examining the statistics of individually noise-dominated images.  This "intensity mapping" technique has garnered significant experimental interest in recent years, with a multitude of surveys planned or in progress across the electromagnetic spectrum.  Interpreting the results of these experiments presents a highly complex data science challenge.  I will discuss analytic and computational data analysis techniques we have developed to separate cosmological signals from noise and foreground contamination.  I will further show how intensity maps and conventional surveys such as LSST can be combined to yield much more than the sum of their parts  With these new analyses in hand, I will demonstrate that we can use them to open unique new windows into cosmological mysteries such as the cosmic star formation history, the interaction between supermassive black holes and their host galaxies, the nature of dark energy, and the birth of the universe itself.