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LPAs (Leavitt Path Algebras) were defined recently (Abrams and Aranda Pino, 2005; Ara, Moreno and Pardo, 2007) but they have roots in the works of Leavitt in the 60s focused on understanding the extent of the failure of the IBN (Invariant Basis Number) property for arbitrary rings. A ring has IBN if any two bases of a finitely generated free module have the same number of elements. Fields, division rings, commutative rings, Noetherian rings all have IBN. However, the rings L(1,n) defined by Leavitt (1962) and their analytic cousins the C*-algebras of Cuntz (1977) are not artificial and pathological structures constructed only for the sake of providing counter examples; for instance, they implicitly come up in Signal Processing (as the algebras generated by the downsampling and upsampling operators). Moreover Leavitt's work (1962, 1965) provided important impetus for major developments in non- commutative ring theory in the 1970s by Cohn, Bergman and others. I plan to start with the basic definitions, state some fundamental results, explain the criterion for an LPA to have IBN (joint work with Muge Kanuni Er) and indicate the ideas involved in the recent classification of the finite dimensional representations (jointly with Ayten Koc). LPAs are (Cohn) localizations of Path (or Quiver) Algebras whose finite dimensional representations are usually wild, but the category of finite dimensional representations of LPAs turn out to be tame with a very reasonable classification of all the indecomposables and the simples. All finite dimensional quotients of LPAs are also easy to describe.