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Realization of the conformal higher spin symmetry on the 4d massless fieldsupermultiplets is given. The self-conjugated supermultiplets, including thelinearized ${\cal N}=4$ SYM theory, are considered in some detail. Dualitybetween non-unitary field-theoretical representations and the unitarydoubleton--type representations of the 4d conformal algebra $su(2,2)$ isformulated in terms of a Bogolyubov transform. The set of 4d massless fields ofall spins is shown to form a representation of $sp(8)$.  The obtained results are extended to the generalized superspace invariantunder $osp(L, 2M)$ supersymmetries. World line particle interpretation of thefree higher spin theories in the $osp(2\N, 2M)$ invariant (super)space isgiven. Compatible with unitarity free equations of motion in the $osp(L,2M)$invariant (super)space are formulated. A conjecture on the chain of$AdS_{d+1}/CFT_d \to AdS_{d}/CFT_{d-1} \to ...$ dualities in the higher spingauge theories is proposed.Comment: Latex, 63 pages; no figures.V2 Typos corrected, References updated.  V3.Typos correced, references and an explanation on the relationship with  previous work in section 4 are added. The version to appear in Phys.Rev.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

Conformal higher spin symmetries of 4D massless supermultiplets and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>osp</mml:mi><mml:mo>(</mml:mo><mml:mi>L</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:mo>)</mml:mo><mml:mn /></mml:math>invariant equations in generalized (super)space

Conformal higher spin symmetries of 4D massless supermultiplets andosp(L,2M)invariant equations in generalized (super)space