  We discuss hybrid Monte Carlo algorithms for odd-flavor lattice QCD
simulations. The algorithms include a polynomial approximation which enables us
to simulate odd-flavor QCD in the framework of the hybrid Monte Carlo
algorithm. In order to make the algorithms exact, the correction factor to the
polynomial approximation is also included in an economical, stochastic way. We
test the algorithms for $n_f=1$, 1+1 and 2+1 flavors and compare results with
other algorithms.
