  In this note, we review some recent developments related to metric aspects of
scalar curvature from the point of view of index theory for Dirac operators. In
particular, we revisit index-theoretic approaches to a conjecture of Gromov on
the width of Riemannian bands $M \times [-1,1]$, and on a conjecture of
Rosenberg and Stolz on the non-existence of complete positive scalar curvature
metrics on $M \times {\mathbb R}$. We show that there is a more general
geometric statement underlying both of them implying a quantitative negative
upper bound on the infimum of the scalar curvature of a complete metric on $M
\times {\mathbb R}$ if the scalar curvature is positive in some neighborhood.
We study ($\hat{A}$-)iso-enlargeable spin manifolds and related notions of
width for Riemannian manifolds from an index-theoretic point of view. Finally,
we list some open problems arising in the interplay between index theory,
largeness properties and width.
