  Comparison theorems are foundational to our understanding of the geometric
features implied by various curvature constraints. This paper considers
manifolds with a positive lower bound on either scalar, 2-Ricci, or Ricci
curvature, and contains a variety of theorems which provide sharp relationships
between this bound and notions of {\em{width}}. Some inequalities leverage
geometric quantities such as boundary mean curvature, while others involve
topological conditions in the form of linking requirements or homological
constraints. In several of these results open and incomplete manifolds are
studied, one of which partially addresses a conjecture of Gromov in this
setting. The majority of results are accompanied by rigidity statements which
isolate various model geometries -- both complete and incomplete -- including a
new characterization of round lens spaces, and other models that have not
appeared elsewhere. As a byproduct, we additionally give new and quantitative
proofs of several classical comparison statements such as Bonnet-Myers' and
Frankel's Theorem, as well as a version of Llarull's Theorem and a notable fact
concerning asymptotically flat manifolds. The results that we present vary
significantly in character, however a common theme is present in that the lead
role in each proof is played by \emph{spacetime harmonic functions}, which are
solutions to a certain elliptic equation originally designed to study mass in
mathematical general relativity.
