  A conformal structure on a manifold $M^n$ induces natural second order
conformally invariant operators, called M\"obius and Laplace structures, acting
on specific weight bundles of $M$, provided that $n\ge 3$. By extending the
notions of M\"obius and Laplace structures to the case of surfaces and curves,
we develop here the theory of extrinsic conformal geometry for submanifolds,
find tensorial invariants of a conformal embedding, and use these invariants to
characterize various forms of geodesic submanifolds.
