  We study asymptotic safety of models of the higher derivative quantum gravity
with and without matter. The beta functions are derived by utilizing the
functional renormalization group, and non-trivial fixed points are found. It
turns out that all couplings in gravity sector, namely the cosmological
constant, the Newton constant, and the $R^2$ and $R_{\mu\nu}^2$ coupling
constants, are relevant in case of higher derivative pure gravity. For the
Higgs-Yukawa model non-minimal coupled with higher derivative gravity, we find
a stable fixed point at which the scalar-quartic and the Yukawa coupling
constants become relevant. The relevant Yukawa coupling is crucial to realize
the finite value of the Yukawa coupling constants in the standard model.
