  Toric quiver varieties (moduli spaces of quiver representations) are studied.
Given a quiver and a weight there is an associated quasiprojective toric
variety together with a canonical embedding into projective space. It is shown
that for a quiver with no oriented cycles the homogeneous ideal of this
embedded projective variety is generated by elements of degree at most $3$. In
each fixed dimension $d$ up to isomorphism there are only finitely many
$d$-dimensional toric quiver varieties. A procedure for their classification is
outlined.
