  We construct continuously parametrised families of conformally invariant
boundary operators on densities. These may also be viewed as conformally
covariant boundary operators on functions and generalise to higher orders the
first-order conformal Robin operator and an analogous third-order operator of
Chang-Qing. Our families include operators of critical order on odd-dimensional
boundaries. Combined with the (conformal Laplacian power) GJMS operators, a
suitable selection of the boundary operators yields formally self-adjoint
elliptic conformal boundary problems. Working on a conformal manifold with
boundary, we show that the operators yield odd-order conformally invariant
fractional Laplacian pseudo-differential operators. To do this, we use
higher-order conformally invariant Dirichlet-to-Neumann constructions. We also
find and construct new curvature quantities associated to our new operator
families. These have links to the Branson Q-curvature and include higher-order
generalisations of the mean curvature and the T-curvature of Chang-Qing. In the
case of the standard conformal hemisphere, the boundary operator construction
is particularly simple; the resulting operators provide an elementary
construction of families of symmetry breaking intertwinors between the
spherical principal series representations of the conformal group of the
equator, as studied by Juhl and others. We use our constructions to shed light
on some conjectures of Juhl.
