The growth of structure in our Universe produces rich data that inform our understanding of cosmology and astrophysics. Much of this information is contained in non-Gaussian fluctuations of matter density. I will give an overview of cosmology results by the Dark Energy Survey (DES) that are driven by the weak gravitational lensing signals of these non-Gaussian features. My focus will be on results from the full probability distribution function of matter and galaxy density, and from clusters of galaxies. These features allow key tests of our understanding of cosmology and structure formation, yet pose significant challenges in data processing, computation, and modeling. To fulfill the promise of structure as a decisive test of cosmology, we will have to devise new methods of jointly and fully analyzing the unprecedented data collected but upcoming astronomical surveys.