Let $H=k\cal Q$ be a finite-dimensional connected
wild hereditary path algebra, over some field $k$.
Denote by $H$-reg the category of finite-dimensional
regular $H$-modules, that is, the category of
modules $M$ with $\tau_H^{-m}(\tau_H^m M) \cong M$
for all integers $m$, where $\tau_H$ denotes the
Auslander--Reiten translation. Call a filtration
\begin{equation}
M = M_0\supset M_1\supset\ldots\supset M_r
\supset M_{r+1}=0 \tag{$*$}
\end{equation}
of a regular $H$-module $M$ a {\em regular filtration}
if all subquotients $M_i/M_{i+1}$ are regular. Call
a regular filtration $(*)$ a {\em regular composition
series} if it is strictly decreasing and has no proper
refinement. A regular component $\cal C$ in the
Auslander--Reiten quiver $\Gamma (H)$ of $H$-mod
is called {\em filtration closed} if, for each
$M\in\text{add\,}\cal C$, the additive closure of
$\cal C$, and each regular filtration $(*)$ of $M$,
all the subquotients $M_i/M_{i+1}$ are also in
$\text{add\,}\cal C$. We show that most wild
hereditary algebras have filtration-closed
Auslander--Reiten components. Moreover,
we deduce from this that there are also {\em almost
serial} components, that is regular components
$\cal C$, such that any indecomposable $X\in\cal C$
has a unique regular composition series.
This composition series coincides with the
Auslander--Reiten filtration of $X$, given by the
maximal chain of irreducible monos ending at $X$. 1991 Mathematics Subject Classification:
16G70, 16G20, 16G60, 16E30.