  We construct gauge theory of interacting symmetric traceless tensor fields of
all ranks s=0,1,2,3, ... which generalizes Weyl-invariant dilaton gravity to
the higher spin case, in any dimension d>2. The action is given by the trace of
the projector to the subspace with positive eigenvalues of an arbitrary
hermitian differential operator H, the symmetric tensor fields emerge after
expansion of the latter in power series in derivatives. After decomposition in
perturbative series around a conformally flat point H=\Box, the quadratic part
of the action breaks up as a sum of free gauge theories of symmetric traceless
tensors of rank s with actions of d-4+2s order in derivatives introduced in 4d
case by Fradkin and Tseytlin and studied at the cubic order level by Fradkin
and Linetsky. Higher orders in interaction are well-defined. We discuss in
detail global symmetries of the model which give rise to infinite dimensional
conformal higher spin algebras for any d. We stress geometric origin of
conformal higher spin fields as background fields of a quantum point particle,
and make the conjecture generalizing this geometry to the system "tensionless
d-1 brane + Fronsdal higher spin massless fields in d+1 dimensions". We propose
a candidate on the role of Higgs-like higher spin compensator able to
spontaneously break higher spin symmetries. At last, we make the conjecture
that, in even dimensions d, the action of conformal higher spin theory equals
the logarithmically divergent term of the action of massless higher spin fields
on AdS_{d+1} evaluated on the solutions of Dirichlet-like problem, where
conformal higher spin fields are boundary values of massless higher spin fields
on AdS_{d+1}, the latter conjecture provides information on the full higher
spin action in AdS_{d+1}.
