Cluster algebras are certain commutative rings (whose  
generators are produced combinatorially) which lie somewhere between a  
polynomial ring and its field of fractions.  They were introduced in  
2001 by Fomin and Zelevinsky for studying total positivity and dual  
canonical bases in semisimple algebraic groups.  Since then, they have  
seen relations to categorification, string theory, Teichmuller theory,  
representation theory, algebraic geometry, and more.

I will give an introduction to the theory of cluster algebras,  
illustrated with many examples.  Part of this talk will be based on  
joint work with Greg Muller.  There are many examples of cluster  
algebras arising in nature (as coordinate rings of varieties appearing  
in representation theory), some of which will be given in Maitreyee  
Kulkarni's talk, which is directly after my talk.