Abstract: In profoundly influential works, Shannon and Huffman showed that if Alice wants to send a message X to Bob, it's sufficient for her to send roughly H(X) bits, where H denotes Shannon's entropy function. This means that every message can be compressed to its information content. Can one prove similar results in the interactive setting, where Alice and Bob engage in an interactive communication protocol? The standard way to formalize this question is by asking whether the communication complexity of a communication task is always (roughly) bounded by its (internal or external) information complexity. In this talk, we will survey known compression protocols and see that for some interesting special cases, interactive compression is possible (e.g. [Kol15]). We then sketch the proof of [GKR15], showing that interactive compression is impossible in general, by presenting a communication task whose communication complexity is exponentially large in its (external) information cost.