  Asymptotic integration theory gives a collection of results which provide a
thorough description of the asymptotic growth and zero distribution of
solutions of (*) $f''+P(z)f=~0$, where $P(z)$ is a polynomial. These results
have been used by several authors to find interesting properties of solutions
of (*). That said, many people have remarked that the proofs and discussion
concerning asymptotic integration theory that are, for example, in E.~Hille's
1969 book \emph{Lectures on Ordinary Differential Equations} are difficult to
follow. The main purpose of this paper is to make this theory more
understandable and accessible by giving complete explanations of the reasoning
used to prove the theory and by writing full and clear statements of the
results. A considerable part of the presentation and explanation of the
material is different from that in Hille's book.
