In the middle part of the last century, Alexander Grothendieck introduced a number of revolutionary concepts into algebraic geometry. This text for advanced graduate students and beginning researchers offers detailed treatments of the theories and proofs he presented in the collection titled Fondements de la Geometrie Algebrique — commonly referred to as FGA. Four main themes are addressed: descent theory; Hilbert and Quot schemes; the formal existence theorem; and the Picard scheme. The text is based upon course materials developed for a summer school held in 2003 at the International Center for Theoretical Physics in Trieste, Italy.