The goal of percolation theory is to study the geometric properties of random networks. In this talk I will describe two variants of percolation: 1. a static variant, called “ordinary percolation”; 2. a dynamic variant, called “invasion percolation”. My main target is to show that these two variants exhibit remarkable similarities and remarkable differences. Ordinary percolation is used to model the connectivity in a random network (“What is connected to what?”). Invasion percolation is used to model the spread of a virus through a random network (“What gets infected?”). I will present main results, open questions and key challenges. No prior knowledge of percolation is necessary for the talk.