Understanding the spectral norm of random matrices is a problem of basic interest in several areas of pure mathematics (probability theory, functional analysis, combinatorics) and in applied mathematics, statistics, and computer science. While the spectral norm of classical random matrix models is well understood, existing methods almost always fail to be sharp in the presence of nontrivial structure. In this talk, I will discuss new bounds on the norm of random matrices with independent entries that are sharp under mild conditions. These bounds shed significant light on the nature of the problem, and make it possible to effortlessly address otherwise nontrivial problems such as identifying the phase transition of the spectral edge of random band matrices.