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.