  In this survey, we discuss various aspects of the minimal surface equation in
the three-sphere S^3. After recalling the basic definitions, we describe a
family of immersed minimal tori with rotational symmetry. We then review the
known examples of embedded minimal surfaces in S^3. Besides the equator and the
Clifford torus, these include the Lawson and Kapouleas-Yang examples, as well
as a new family of examples found recently by Choe and Soret. We next discuss
uniqueness theorems for minimal surfaces in S^3, such as the work of Almgren on
the genus 0 case, and our recent solution of Lawson's conjecture for embedded
minimal surfaces of genus 1. More generally, we show that any minimal surface
of genus 1 which is Alexandrov immersed must be rotationally symmetric. We also
discuss Urbano's estimate for the Morse index of an embedded minimal surface
and give an outline of the recent proof of the Willmore conjecture by Marques
and Neves. Finally, we describe estimates for the first eigenvalue of the
Laplacian on a minimal surface.
