The physical limitations of omni-directional antennas are considered. With the use of the spherical wave functions to describe the field, the directivity gain G and the Q of an unspecified antenna are calculated under idealized conditions. To obtain the optimum performance, three criteria are used, (1) maximum gain for a given complexity of the antenna structure, (2) minimum Q, (3) maximum ratio of G/Q. It is found that an antenna of which the maximum dimension is 2a has the potentiality of a broad band width provided that the gain is equal to or less than 4a/λ. To obtain a gain higher than this value, the Q of the antenna increases at an astronomical rate. The antenna which has potentially the broadest band width of all omni-directional antennas is one which has a radiation pattern corresponding to that of an infinitesimally small dipole.