  We make classifications of gradient Ricci solitons $(M, g, f)$ with harmonic
Weyl curvature. As a local classification, we prove that the soliton metric $g$
is locally isometric to one of the following four types: an Einstein manifold,
the Riemannian product of a Ricci flat manifold and an Einstein manifold, a
warped product of $\mathbb{R}$ and an Einstein manifold, and a singular warped
product of $\mathbb{R}^2$ and a Ricci flat manifold. Compared with the previous
four-dimensional study in \cite{Ki}, we have developed a novel method of {\it
refined adapted frame fields} and overcome the main difficulty arising from a
large number of Riemmannian connection components in dimension$ \geq 5$.
  Next we have obtained a classification of {\it complete} gradient Ricci
solitons with harmonic Weyl curvature. For the proof, using the real analytic
nature of $g$ and $f$, we elaborate geometric arguments to fit together local
regions.
