  Here we survey the compactness and geometric stability conjectures formulated
by the participants at the 2018 IAS Emerging Topics Workshop on {\em Scalar
Curvature and Convergence}. We have tried to survey all the progress towards
these conjectures as well as related examples, although it is impossible to
cover everything. We focus primarily on sequences of compact Riemannian
manifolds with nonnegative scalar curvature and their limit spaces. Christina
Sormani is grateful to have had the opportunity to write up our ideas and has
done her best to credit everyone involved within the paper even though she is
the only author listed above. In truth we are a team of over thirty people
working together and apart on these deep questions and we welcome everyone who
is interested in these conjectures to join us.
