  We investigate the properties of $Q$-balls in $d$ spatial dimensions. First,
a generalized virial relation for these objects is obtained. We then focus on
potentials $V(\phi\phi^{\dagger})= \sum_{n=1}^{3} a_n(\phi\phi^{\dagger})^n$,
where $a_n$ is a constant and $n$ is an integer, obtaining variational
estimates for their energies for arbitrary charge $Q$. These analytical
estimates are contrasted with numerical results and their accuracy evaluated.
Based on the results, we offer a simple criterion to classify ``large'' and
``small'' $d$-dimensional $Q$-balls for this class of potentials. A minimum
charge is then computed and its dependence on spatial dimensionality is shown
to scale as $Q_{\rm min} \sim \exp(d)$. We also briefly investigate the
existence of $Q$-clouds in $d$ dimensions.

High Energy Physics - Theory