Given a field K and a directed graph E, one can construct a K-algebra L_K(E) that generalizes the Leavitt algebras introduced in the 1950's. These Leavittt path algebras are defined in a way similar to that of graph C^*-algebras, and surprisingly it has been found that many similar results hold for the two classes (although the proofs for each class have been different). We will discuss some fundamental structure theorems for the Leavitt path algebras and discuss how these theorems give insight into the relationship between Leavitt path algebras and graph C^*-algebras.