Normal cells become malignant and metastatic through the acquisition of DNA variants. If the genome is a book, the simplest and most numerous mutations comprise small “typos” in individual words. However, the most difficult variants to detect (and interpret) are the result of copying, deletion, and rearrangement of entire paragraphs and chapters. Some of these “structural variants” are known to bring together genes or regulatory sequences that drive cancer progression and render tumors sensitive to drugs. However, our understanding of the evolution and impact of most cancer structural variants is still rudimentary. A key technical obstacle for comprehending this structure is to see beyond the tiny (or sometimes less tiny) shreds of sequence that constitute our data. A deeper and more conceptual challenge is how to represent this altered structure (or our best guess of it) in a form that will allow exploration and statistical mining of recurrent and biologically meaningful patterns. I’ll review state of the art approaches towards understanding complex structural variation in cancer, and introduce the paradigm of a cancer graph genome. I’ll describe approaches to visualizing and mining cancer genome graphs to detect recurrent patterns (mutational signatures) and interpret complex mutational events.