  This paper is an expanded version of two talks given by the author at the
Summer School on the Interactions between Homotopy Theory and Algebra at the
University of Chicago, July 26 to August 6, 2004. It describes a connection
between model categories, a structure invented by algebraic topologists that
allows one to introduce the ideas of homotopy theory to situations far removed
from topological spaces, and cotorsion pairs, an algebraic notion that
simultaneously generalizes the notion of projective and injective objects. It
also gives some applications of this connection, some due to the authour about
model structures on Z[G]-modules for G a finite group, and some due to Jim
Gillespie about flat model structures of sheaves.
