  Seiberg-Witten theory leads to a delicate interplay between Riemannian
geometry and smooth topology in dimension four. In particular, the scalar
curvature of any metric must satisfy certain non-trivial estimates if the
manifold in question has a non-trivial Seiberg-Witten invariant. However, it
has recently been discovered that similar statements also apply to other parts
of the curvature tensor. This article presents the most salient aspects of
these curvature estimates in a self-contained manner, and shows how they can be
applied to the theory of Einstein manifolds.
  We then probe the issue of whether the known estimates are optimal by
relating this question to a certain conjecture in Kaehler geometry.
