  This is an expository account of Grothendieck's construction of Hilbert and
Quot Schemes, following his talk `Techniques de construction et theoremes
d'existence en geometrie algebriques IV : les schemas de Hilbert', Seminaire
Bourbaki 221 (1960/61), together with further developments by Mumford and by
Altman and Kleiman. Hilbert and Quot schemes are fundamental to modern
Algebraic Geometry, in particular, for deformation theory and moduli
constructions. These notes are based on a series of six lectures in the summer
school `Advanced Basic Algebraic Geometry', held at the Abdus Salam
International Centre for Theoretical Physics, Trieste, in July 2003.
