  We prove existence of isoperimetric regions for every volume in non-compact
Riemannian $n$-manifolds $(M,g)$, $n\geq 2$, having Ricci curvature $Ric_g\geq
(n-1) k_0 g$ and being locally asymptotic to the simply connected space form of
constant sectional curvature $k_0$; moreover in case $k_0=0$ we show that the
isoperimetric regions are indecomposable. We also discuss some physically and
geometrically relevant examples. Finally, under assumptions on the scalar
curvature we prove existence of isoperimetric regions of small volume.
