Classical field theories of scalars that are invariant under a U(1) symmetry can have non-perturbative soliton solutions called Q-balls: spherical bound states that are the energy minimum for a fixed U(1) charge Q. The underlying non-linear differential equations cannot be solved analytically, but I will present analytical approximations that work remarkably well and allow us to understand both ground and excited Q-ball states with ease. The global U(1) symmetry can also be promoted to a gauge symmetry, leading to gauged Q-balls and Q-shells; once again we can find exceptionally good analytic approximations to describe these objects and their stability.