  Bosons carrying a conserved charge can form stable bound states if their
Lagrangian contains attractive self-interactions. Bound-state configurations
with a large charge $Q$ can be described classically and are denoted as
Q-balls, their properties encoded in a non-linear differential equation. Here,
we study Q-balls in arbitrary polynomial single-scalar-field potentials both
numerically and via various analytical approximations. We highlight some
surprising universal features of Q-balls that barely depend on the details of
the potential. The polynomial potentials studied here can be realized in
renormalizable models involving additional heavy or light scalars, as we
illustrate with several examples.
