  For conformal geometries of Riemannian signature, we provide a comprehensive
and explicit treatment of the core local theory for embedded submanifolds of
arbitrary dimension. This is based in the conformal tractor calculus and
includes a conformally invariant Gauss formula leading to conformal versions of
the Gauss, Codazzi, and Ricci equations. It provides the tools for
proliferating submanifold conformal invariants, as well for extending to
conformally singular Riemannian manifolds the notions of mean curvature and of
minimal and CMC submanifolds. A notion of distinguished submanifold is defined
by asking the tractor second fundamental form to vanish. We show that for the
case of curves this exactly characterises conformal geodesics (a.k.a. conformal
circles) while for hypersurfaces it is the totally umbilic condition. So, for
other codimensions, this unifying notion interpolates between these extremes,
and we prove that in all dimensions this coincides with the submanifold being
weakly conformally circular, meaning that ambient conformal circles remain in
the submanifold. Stronger notions of conformal circularity are then
characterised similarly. Next we provide a very general theory and construction
of quantities that are necessarily conserved along distinguished submanifolds.
This first integral theory thus vastly generalises the results available for
conformal circles in [56]. We prove that any normal solution to an equation
from the class of first BGG equations can yield such a conserved quantity, and
show that it is easy to provide explicit formulae for these. Finally we prove
that the property of being distinguished is also captured by a type of moving
incidence relation. This second characterisation is used to show that, for
suitable solutions of conformal Killing-Yano equations, a certain zero locus of
the solution is necessarily a distinguished submanifold.
