  We present the matter power spectrum in redshift space including two-loop
corrections. We follow a strictly perturbative approach incorporating all
non-linearities entering both via the redshift-space mapping and within real
space up to the required (fifth) order, complemented by suitable effective
field theory (EFT) corrections. This approach can a priori be viable up to
scales of order $0.2h~\mathrm{Mpc}^{-1}$ beyond which power suppression related
to the finger-of-God effect becomes non-perturbatively strong. We extend a
simplified treatment of EFT corrections at two-loop order from real to redshift
space, making sure that the leading UV-sensitivity of both the single-hard and
double-hard limit of the two-loop contributions to the power spectrum is
accounted for, and featuring two free parameters for each multipole. Taking
also infrared-resummation into account, we calibrate with and compare to
Quijote $N$-body simulations for the monopole and quadrupole at redshifts $z=0$
and $z=0.5$. We find agreement within sample variance (at percent-level) up to
$0.18h~\mathrm{Mpc}^{-1}$ at two-loop order, compared to
$0.1h~\mathrm{Mpc}^{-1}$ at one-loop. We also investigate the role of
higher-derivative corrections.
