The equation

y^3 = x^6 + 23x^5 + 37x^4 + 691x^3 − 631204x^2 + 5169373941

obviously has the solution (1, 1729). Are there any other solutions in rational numbers? The study of integral or rational solutions to polynomial equations, sometimes known as the theory of Diophantine equations, is among the oldest pursuits in mathematics. This lecture will give an idiosyncratic survey of the remarkable advances made in the 20th and 21st century for the special case of equations of two variables. The emphasis will be on the techniques of arithmetic geometry, the study of spaces built up of finitely-generated systems of numbers.