S. T. Yau conjectured in the 80's that every compact Riemannian 3-manifold should contain infinitely many different minimal surfaces (smooth, closed). This was proved last year by Antoine Song.

Song built on a long story of breakthroughs in the area by Fernando Marques and Andre Neves using Min-Max theory. Another earlier big result was the solution of the Willmore Conjecture about embedding minimal tori: The Willmore energy $\int_{\Sigma}H^2$ of any smoothly immersed torus in $\mathbf{R}^3$ is at least $2\pi^2$.