Building on the introduction to spin geometry taught by Bernhard Hanke, I will use spinorial methods to investigate manifolds whose scalar curvature satisfies restrictions. Scalar curvature contains much less information than sectional or Ricci curvature which is why relatively sophisticated methods are needed to understand its implications. The results to be covered include Geroch’s conjecture on the impossibility of positive scalar curvature on tori as well as Llarull’s famous rigidity theorem.