In the first billion years after the Big Bang, the universe's hydrogen gas became ionized, an event known as reionization. Reionization represents a fundamental transition in the universe's properties, and yet we know little about how it occurred. The most likely explanation is that ionizing, Lyman continuum (LyC) photons escaped into the intergalactic medium from early star-forming galaxies. However, most star-forming galaxies show no sign of LyC escape. If reionization was caused by galaxies, which galaxies were responsible? The recent discovery of escaping ionizing radiation from the unusual "Green Pea" galaxies has provided new clues to this puzzle. I will discuss what we are learning from the Green Peas about how ionizing radiation escapes galaxies and about the possible properties of the galaxies that reionized the universe.