  We review analytic methods to perform the post-Newtonian expansion of
gravitational waves induced by a particle orbiting a massive compact body,
based on the black hole perturbation theory. There exist two different methods
of the post-Newtonian expansion. Both are based the Teukolsky equation. In one
method, the Teukolsky equation is transformed into a Regge-Wheeler type
equation that reduces to the standard Klein-Gordon equation in the flat space
limit, while in the other method, which were introduced by Mano, Suzuki and
Takasugi relatively recently, the Teukolsky equation is directly used in its
original form. The former has an advantage that it is intuitively easy to
understand how various curved space effects come into play. However, it becomes
increasingly complicated when one goes on to higher and higher post-Newtonian
orders. In contrast, the latter has an advantage that a systematic calculation
to higher post-Newtonian orders is relatively easily implementable, but
otherwise so mathematical that it is hard to understand the interplay of higher
order terms. In this paper, we review both methods so that their pros and cons
may be clearly seen. We also review some results of calculations of
gravitational radiation emitted by a particle orbiting a black hole.
