We will present the enumerative min-max theory, which relates the number of genus g minimal surfaces in 3-manifolds to topological properties of the set of all embedded surfaces of genus ≤g. As a consequence, we can show that in every 3-sphere of positive Ricci curvature, there exist ≥5 minimal tori (confirming a conjecture by B. White (1989) in the Ricci-positive case), ≥4 minimal surfaces of genus 2, and ≥1 minimal surface of genus g for all g. This is based on a joint work with Yangyang Li and Zhihan Wang.