At an early age, we learn about phase transitions: at what temperature does solid turn to liquid and liquid turn to gas? The phenomenon of phase transitions occurs both in nature and in mathematics. For example, suppose we create a random graph by flipping independent coins to determine if vertices are connected. If each coin is biased, what is the typical structure of the network? How does this change as we adjust the bias of the coin, and is there a “threshold” at which the structure changes dramatically? In this talk, we'll explore several notions of such thresholds, including those defined by structural properties, by the existence of efficient sampling algorithms, and by the evolution rate of a dynamical process. Our examples will come from combinatorics, computer science, and statistical physics, but no background with these topics will be assumed!