Understanding BBN: the physics and its history

Big-bang nucleosynthesis (BBN), today a pillar of modern cosmology, began with the trailblazing 1948 paper of Alpher, Bethe and Gamow. In it, they proposed non-equilibrium nuclear processes in the early Universe ($t \sim 1000\,$sec) and an early radiation-dominated phase to explain the abundances of all the chemical elements. Their model was fundamentally flawed, but initiated a complex and interesting path to the modern theory of BBN, which explains only the abundances of the lightest chemical elements (mostly $^4$He) and the discovery of the cosmic microwave background (CMB). The purpose of this paper is to clarify the basic physics of BBN, adding some new insights, and to describe how the modern theory developed. I finish with a discussion of two misunderstandings about BBN that still persist and the tale of the pre-discovery predictions of the temperature of the CMB and the missed opportunity it turned out to be.