The contribution of the electromagnetic self-field to the energy and momentum of a charge and/or current carrying body is considered within classical physics and discussed in simple terms using nontrivial exactly solvable examples. The Lorentz-transformation properties of the energy and momentum of the body can be maintained by taking into account the relativistic effects of the mechanical stresses due the sources of the self-field, or by separating the energy and momentum of a moving self-field from those of a nonelectromagnetic origin in a relativistically covariant way. In either procedure, a proper account must be taken of the hidden mechanical momentum that, in general, is contained in a stationary body that carries both charge and current, and which is equal and opposite to the momentum of the static electromagnetic self-field. Hidden momentum and the momentum of a static electromagnetic field are indispensable concepts of direct physical significance in classical electrodynamics.