Explore the fascinating world of Diophantine equations in this 53-minute American Mathematical Society Invited Address from the 2020 Joint Mathematics Meetings. Delve into the geometric aspects of these equations as Anthony Várilly-Alvarado from Rice University guides you through topics such as the sum of three cubes, local obstructions, magic squares, and the projective plane. Engage with audience participation segments and learn about key players in the field, rational curves, K3 surfaces, and systems of equations. Conclude with an exploration of the Four Squares Theorem and participate in a question-and-answer session to deepen your understanding of this complex mathematical subject.