  Free Lagrangians are found both for gauge and non-gauge bosonic conformal
fields of any symmetry type and in any space-time dimension. Conformal gauge
fields of various types, their gauge transformations and gauge invariant field
strengths (generalized Weyl tensors), which are derived by the $\sigma_-$
cohomology technics in the frame-like formulation, are shown to correspond to
supersymmetric vacua of certain supersymmetric matrix mechanics. The
correspondence between conformal and $AdS_d$ higher-spin models, that turn out
to have identical generalized Weyl tensors, is discussed.
