  The rare decay B -> K* nu anti-nu allows a transparent study of Z penguin and
other electroweak penguin effects in New Physics (NP) scenarios in the absence
of dipole operator contributions and Higgs (scalar) penguin contributions that
are often more important than Z contributions in B -> K* l+l- and B(s) -> l+l-
decays. We present a new analysis of B -> K* nu anti-nu with improved form
factors and of the decays B -> K nu anti-nu and B -> X(s) nu anti-nu in the SM
and in a number of NP scenarios like the general MSSM, general scenarios with
modified Z/Z' penguins and in a singlet scalar extension of the SM. We also
summarize the results in the Littlest Higgs model with T-parity and a
Randall-Sundrum (RS) model with custodial protection of left-handed Z-di-dj
couplings. Our SM prediction BR(B -> K* nu anti-nu)=(6.8^+1.0_-1.1) x 10^-6
turns out to be significantly lower than the ones present in the literature.
Our improved calculation BR(B -> X(s) nu anti-nu)=(2.7+-0.2) x 10^-5 in the SM
avoids the normalization to the BR(B -> X(c) e anti-nu(e)) and, with less than
10% total uncertainty, is the most accurate to date. The results for the SM and
NP scenarios can be transparently summarized in a (epsilon,eta) plane with a
non-vanishing eta signalling the presence of new right-handed down-quark
flavour violating couplings which can be ideally probed by the decays in
question. Measuring the three branching ratios and one additional polarization
observable in B -> K* nu anti-nu allows to overconstrain the resulting point in
the (epsilon,eta) plane with (epsilon,eta)=(1,0) corresponding to the SM. The
correlations of these three channels with the rare decays K+ -> pi+ nu anti-nu,
KL -> pi0 nu anti-nu, B -> X(s) l+ l- and B(s) -> mu+ mu- offer powerful tests
of New Physics with new right-handed couplings and non-MFV interactions.
