In this paper the center of a Leavitt path algebra is computed for a wide range of situations. A basis as a K -vector space is found for Z(L(E)) when L(E) enjoys some finiteness condition such as being artinian, semisimple, noetherian and locally noetherian. The main result of the paper states that a simple Leavitt path algebra L(E) is central (i.e. the center reduces to the base field K ) when L(E) is unital and has zero center otherwise. Finally, this result is extended, under some mild conditions, to the case of exchange Leavitt path algebras.