  This article is based on my lecture notes from summer schools at the
Universities of Utah (June 2007) and Warwick (September 2007). We provide an
introduction to explicit methods in the study of moduli spaces of quiver
representations and derived categories arising in toric geometry. The first
main goal is to present the noncommutative geometric approach to semiprojective
toric varieties via quivers. To achieve this, we use geometric invariant theory
to construct both semiprojective toric varieties and moduli spaces of quiver
representations. The second main goal builds on the first by presenting an
introduction to explicit methods in derived categories of coherent sheaves in
toric geometry. We recall the notion of tilting bundles with examples, and
describe the McKay correspondence as a derived equivalence in some detail
following Bridgeland, King and Reid. We also describe extensions of their
result beyond the $G$-Hilbert scheme to other fine moduli spaces of bound
quiver representations.
