  We have studied two-photon transitions from Upsilon(3S) decays recorded by
the CLEO-III detector in exclusive events with two photons and either two
electrons or two muons. We obtain precision measurements of the chi_b(2P_J)
masses for J=2 and J=1. The transition rates for all three spin states of the
2P triplet are measured with improved precision which leads to a better
determination of their hadronic width ratios. We also observe rare transitions
via the chi_b(1P) states. The measured rates for these transitions allow a
determination of <1P|r|3S>, the E1 matrix element, which is more sensitive to
the structure of the bb states than the <2P|r|3S> matrix element which
dominates the radiative decays of the Upsilon(3S) state. We also present first
upper limits on the branching ratios for Upsilon(3S)-> pi0 Upsilon(2S),
Upsilon(3S) -> pi0 Upsilon(1S) and a new limit for the branching ratio for
Upsilon(3S) -> eta Upsilon(1S).
