We give a review of the one-loop divergences in higher derivative gravity
theories. We first make the bilinear expansion in the quantum fluctuation on
arbitrary backgrounds, introduce a higher-derivative gauge fixing and show that
higher-derivative gauge fixing must have ghosts in addition to those naively
expected. We give general formulae for the one-loop divergences in such
theories, and give explicit results for theories with quadratic curvature
terms. In this calculation, we need the heat kernel coefficients for the
four-derivative minimal operators and two-derivative nonminimal vector
operators, which are summarized. We also discuss the beta functions in the
renormalization group, and show that the dimensionless couplings are
asymptotically free. The calculation is also extended to the theories with
arbitrary functions of $R$ and $R_{\mu\nu}^2$. We show that the result is
independent of metric parametrization and gauge on shell.
