A difficulty with present mathematical antenna models is the fact that there is no theoretical justification for equating the computed input impedances with those measured experimentally. It is shown that a rigorous definition of antenna impedance can be based only on a computation of the complex power in the electromagnetic field from energy integrals. To account for this power it is essential to consider the transmission-line feed to the antenna, and an antenna model is derived on this basis. This model is then compared with presently accepted models. The usefulness of the induced-e.m.f. method is discussed. It is then shown that the slice generator used in the Hallen-King model does not satisfy the requirement for the conservation of power and that the impedance determined from this model depends on a semi-empirical choice of expansion parameter.