In the era of big data, the significant growth in graph size renders numerous traditional algorithms, including those with polynomial or even linear time complexity, inefficient. Therefore, we need novel approaches for efficiently processing massive graphs. In this talk, I will discuss two modern approaches towards this goal: structure exploitation and graph compression. I will first show how to utilize graph structure to design better approximation algorithms, showcasing my work on the Graph Crossing Number problem. I will then show how to compress massive graphs into smaller ones while preserving their flow/cut/distance structures and thereby obtaining faster algorithms.