Most of the rings one encounters as 'basic examples' satisfy the 'Invariant Basis Number' property: for every pair of positive integers m and n, if the free left R-modules _RR^(m) and _RR⁽ⁿ⁾ are isomorphic, then m=n. There are, however, many important classes of rings which do not have this property. While at first glance such rings might seem pathological, in fact they arise quite naturally in a number of contexts (e.g. as endomorphism rings of infinite dimensional vector spaces), and possess a significant (perhaps surprising) amount of structure. In this presentation we describe a class of such rings, the (now-classical) Leavitt algebras, and then describe their recently developed generalizations, the Leavitt path algebras. One of the nice aspects of this subject is that pictorial representations (using graphs) of the algebras are readily available. In addition, there are strong connections between these algebraic structures and a class of C*-algebras (the 'Cuntz-Krieger graph C*-algebras'), a connection which is currently the subject of great interest to both algebraists and analysts.