  We prove that the scaling limit of loop-erased random walk in a simply
connected domain $D$ is equal to the radial SLE(2) path in $D$. In particular,
the limit exists and is conformally invariant. It follows that the scaling
limit of the uniform spanning tree in a Jordan domain exists and is conformally
invariant. Assuming that the boundary of the domain is a $C^1$ simple closed
curve, the same method is applied to show that the scaling limit of the uniform
spanning tree Peano curve, where the tree is wired along a proper arc $A$ on
the boundary, is the chordal SLE(8) path in the closure of $D$ joining the
endpoints of $A$. A by-product of this result is that SLE(8) is almost surely
generated by a continuous path. The results and proofs are not restricted to a
particular choice of lattice.
