  Moduli spaces of sheaves and Hilbert schemes of points have experienced a
recent resurgence in interest in the past several years, due largely to new
techniques arising from Bridgeland stability conditions and derived category
methods. In particular, classical questions about the birational geometry of
these spaces can be answered by using new tools such as the positivity lemma of
Bayer and Macr\`i. In this article we first survey classical results on moduli
spaces of sheaves and their birational geometry. We then discuss the
relationship between these classical results and the new techniques coming from
Bridgeland stability, and discuss how cones of ample divisors on these spaces
can be computed with these new methods. This survey expands upon the author's
talk at the 2015 Bootcamp in Algebraic Geometry preceding the 2015 AMS Summer
Research Institute on Algebraic Geometry at the University of Utah.
