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The first half of this talk will be an introduction to LPAs (Leavitt Path Algebras), in particular their representations. In the second half of the talk I'll try to indicate the recent classification of the simple modules when the LPA has polynomial growth (joint work with Murad Özaydin). The interesting class of Leavitt path algebras of polynomial growth (i.e. finite Gelfand-Kirillov dimension) include the algebraic Toeplitz/Jacobson algebra and algebraic quantum spheres of every dimension. All the relevant terminology will be dened and explained, the talk is aimed at graduate students.