  Let $(M^n,g)$ be a complete Riemannian manifold which is not isometric to
$\mathbb{R}^n$, has nonnegative Ricci curvature, Euclidean volume growth, and
quadratic Riemann curvature decay. We prove that there is a set
$\mathcal{G}\subset (0,\infty)$ with density $1$ at infinity such that for
every $V\in \mathcal{G}$ there is a unique isoperimetric set of volume $V$ in
$M$, and its boundary is strictly volume preserving stable.
  The latter result cannot be improved to uniqueness or strict stability for
every large volume. Indeed, we construct a complete Riemannian surface that
satisfies the previous assumptions, and with the following property: there are
arbitrarily large and diverging intervals $I_n\subset (0,\infty)$ such that
isoperimetric sets with volumes $V\in I_n$ exist, but they are neither unique
nor they have strictly volume preserving stable boundaries.
  The proof relies on a set of new ideas, as the present setting goes beyond
the range of applicability of the methods based on the implicit function
theorem, and no symmetry is assumed.
