The discovery of the soliton was one of the great achievements of mathematics from the late 20th century. I will talk about what they are and how they tie in with a deceptively simple equation appearing in geometry and fluid mechanics: the Betchov-Da Rios equation, also know colloquially as the “smoke ring” equation. The equation has rather amazing properties, some of which I will describe. If I have time, I will discuss some more recent applications of the Betchov-Da Rios equation in applied mathematics.