  The conformal equivalence of fourth-order gravity following from a non-linear
Lagrangian L(R) to theories of other types is widely known, here we report on a
new conformal equivalence of these theories to theories of the same type but
with different Lagrangian.
  For a quantization of fourth-order theories one needs a Hamiltonian
formulation of them. One of the possibilities to do so goes back to
Ostrogradski in 1850. Here we present another possibility: A Hamiltonian H
different from Ostrogradski's one is discussed for the Lagrangian L depending
on first and second order drivatives of the position variable q. We add a
suitable divergence to L. Contrary to other approaches no constraint is needed.
One of the canonical equations becomes equivalent to the fourth-order
Euler-Lagrange equation of L.
  Finally, we discuss the stability properties of cosmological models within
fourth-order gravity.
