While entropy was introduced in thermodynamics to describe heat engines, its applications have spread to widely different areas. I will talk about recent research on two such problems. The first is a problem in information theory: how much information can we send over a noisy communication channel, given that the world is described by quantum mechanics? I will explain the so-called "additivity conjecture", which was a proposed way to calculate the communication capacity of such a channel, and I will explain my recent result disproving this conjecture, showing that we can use entanglement to boost communication capacity. The second problem is in quantum systems far from equilbrium. Here I will describe how entropy can arise from quantum entanglement, and I will discuss novel simulation algorithms and future experiments probing the relaxation back to local thermal equilibrium.