Structural studies on living cells by conventional methods are limited to low resolution because radiation damage kills cells long before the necessary dose for high resolution can be delivered. X-ray free-electron lasers circumvent this problem by outrunning key damage processes with an ultra-short and extremely bright coherent X-ray pulse. Diffraction-before-destruction experiments provide high-resolution data from cells that are alive when the femtosecond X-ray pulse traverses the sample. This paper presents two data sets from micron-sized cyanobacteria obtained at the Linac Coherent Light Source, containing a total of 199,000 diffraction patterns. Utilizing this type of diffraction data will require the development of new analysis methods and algorithms for studying structure and structural variability in large populations of cells and to create abstract models. Such studies will allow us to understand living cells and populations of cells in new ways. New X-ray lasers, like the European XFEL, will produce billions of pulses per day, and could open new areas in structural sciences.