  We prove Leavitt path algebra versions of the two uniqueness theorems of
graph C*-algebras. We use these uniqueness theorems to analyze the ideal
structure of Leavitt path algebras and give necessary and sufficient conditions
for their simplicity. We also use these results to give a proof of the fact
that for any graph E the Leavitt path algebra $L_\mathbb{C}(E)$ embeds as a
dense *-subalgebra of the graph C*-algebra C*(E). This embedding has
consequences for graph C*-algebras, and we discuss how we obtain new
information concerning the construction of C*(E).
