  Given a word $w$ and a Parikh vector $\mathcal{P}$, an abelian run of period
$\mathcal{P}$ in $w$ is a maximal occurrence of a substring of $w$ having
abelian period $\mathcal{P}$. Our main result is an online algorithm that,
given a word $w$ of length $n$ over an alphabet of cardinality $\sigma$ and a
Parikh vector $\mathcal{P}$, returns all the abelian runs of period
$\mathcal{P}$ in $w$ in time $O(n)$ and space $O(\sigma+p)$, where $p$ is the
norm of $\mathcal{P}$, i.e., the sum of its components. We also present an
online algorithm that computes all the abelian runs with periods of norm $p$ in
$w$ in time $O(np)$, for any given norm $p$. Finally, we give an $O(n^2)$-time
offline randomized algorithm for computing all the abelian runs of $w$. Its
deterministic counterpart runs in $O(n^2\log\sigma)$ time.
