  We study the exclusive semileptonic $B$-meson decays $B\to
K(\pi)\ell^+\ell^-$, $B\to K(\pi)\nu\bar\nu$, and $B\to\pi\tau\nu$, computing
observables in the Standard model using the recent lattice-QCD results for the
underlying form factors from the Fermilab Lattice and MILC Collaborations.
These processes provide theoretically clean windows into physics beyond the
Standard Model because the hadronic uncertainties are now under good control
for suitably binned observables. For example, the resulting partially
integrated branching fractions for $B\to\pi\mu^+\mu^-$ and $B\to K\mu^+\mu^-$
outside the charmonium resonance region are 1-2$\sigma$ higher than the LHCb
Collaboration's recent measurements, where the theoretical and experimental
errors are commensurate. The combined tension is 1.7$\sigma$. Combining the
Standard-Model rates with LHCb's measurements yields values for the
Cabibbo-Kobayashi-Maskawa (CKM) matrix elements
$|V_{td}|=7.45{(69)}\times10^{-3}$, $|V_{ts}|=35.7(1.5)\times10^{-3}$, and
$|V_{td}/V_{ts}|=0.201{(20)}$, which are compatible with the values obtained
from neutral $B_{(s)}$-meson oscillations and have competitive uncertainties.
Alternatively, taking the CKM matrix elements from unitarity, we constrain
new-physics contributions at the electroweak scale. The constraints on the
Wilson coefficients ${\rm Re}(C_9)$ and ${\rm Re}(C_{10})$ from
$B\to\pi\mu^+\mu^-$ and $B\to K\mu^+\mu^-$ are competitive with those from
$B\to K^* \mu^+\mu^-$, and display a 2.0$\sigma$ tension with the Standard
Model. Our predictions for $B\to K(\pi)\nu\bar\nu$ and $B\to\pi\tau\nu$ are
close to the current experimental limits.
