  We introduce nonlocal minimal surfaces on closed manifolds and establish a
far-reaching Yau-type result: in every closed, $n$-dimensional Riemannian
manifold we construct infinitely many nonlocal $s$-minimal surfaces. We prove
that, when $s\in (0,1)$ is sufficiently close to $1$, the constructed surfaces
are smooth for $n=3$ and $n=4$, while for $n\ge 5$ they are smooth outside of a
closed set of dimension $n-5$. Moreover, we prove surprisingly strong
regularity and rigidity properties of finite Morse index $s$-minimal surfaces
such as a "finite Morse index Bernstein-type result". These properties make
nonlocal minimal surfaces ideal objects on which to apply min-max variational
methods.
