  The gravitational-wave event GW170817 from a binary neutron star merger
together with the electromagnetic counterpart showed that the speed of
gravitational waves $c_t$ is very close to that of light for the redshift
$z<0.009$. This places tight constraints on dark energy models constructed in
the framework of modified gravitational theories. We review models of the
late-time cosmic acceleration in scalar-tensor theories with second-order
equations of motion (dubbed Horndeski theories) by paying particular attention
to the evolution of dark energy equation of state and observables relevant to
the cosmic growth history. We provide a gauge-ready formulation of scalar
perturbations in full Horndeski theories and estimate observables associated
with the evolution of large-scale structures, cosmic microwave background, and
weak lensing by employing a so-called quasi-static approximation for the modes
deep inside the sound horizon.
  In light of the recent observational bound of $c_t$, we also classify
surviving dark energy models into four classes depending on different
structure-formation patterns and discuss how they can be observationally
distinguished from each other. In particular, the nonminimally coupled theories
in which the scalar field $\phi$ has a coupling with the Ricci scalar $R$ of
the form $G_4(\phi) R$, including $f(R)$ gravity, can be tightly constrained
not only from the cosmic expansion and growth histories but also from the
variation of screened gravitational couplings. The cross correlation of
integrated Sachs-Wolfe signal with galaxy distributions can be a key observable
for placing bounds on the relative ratio of cubic Galileon density to total
dark energy density. The dawn of gravitational-wave astronomy will open up a
new window to constrain nonminimally coupled theories further by the modified
luminosity distance of tensor perturbations.
