Information theory started with the problem of sending information over a noisy communication channel. In this talk, I will explain to a general audience the type of tasks one would like to accomplish with a quantum communication channel, either communicating classical or quantum information and using inputs and measurements that are either entangled or not, and I will use entropy differences to quantify the rate at which information can be sent. After motivating the mathematical problems in terms of communication tasks, I will describe recent results in channel capacity, focusing on those aspects that can be understood in terms of probability theory.