Bosonic conformal higher--spin fields of any symmetry

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.