The breakdown of a uniform field is considered to occur by the transition of an electron avalanche proceeding from cathode to anode into a self-propagating streamer, which develops from anode to cathode to form a conducting filament between the electrodes. A criterion is put forward for such a transition, viz., a streamer will develop when the radial field about the positive space charge in an electron avalanche attains a value of the order of the external applied field. For then photoelectrons in the immediate vicinity of the avalanche will be drawn into the stem of the avalanche and will give rise to a conducting filament of plasma, and a self-propagating streamer proceeds towards the cathode. The theory thus depends on ionization by electrons and photo-ionization in the gas and dispenses with the classical assumption of ionization by positive ions in the gas or secondary actions at the cathode. On this basis an equation for breakdown is developed, and reference to $\frac{\ensuremath{\alpha}}{p}\ensuremath{-}\frac{X}{p}$ curves enables the potential required for breakdown to be determined. Satisfactory agreement between calculation and experiment is found in air for values of pressure times gap length down to $p\ensuremath{\delta}\ensuremath{\sim}100$ mm Hg\ifmmode\times\else\texttimes\fi{}cm. The theory does not conform absolutely to Paschen's law, but the deviations are within the present day margins of experimental error. For lower values of $p\ensuremath{\delta}$ the deviation between calculation and experiment is explained by the fact that the density of photo-ionization becomes small, and the secondary mechanism, $\ensuremath{\gamma}$, is observed in this region, so that classical theory applies. The theory is indicated to be consistent with all the requirements so far established in connection with sparks for large $p\ensuremath{\delta}$ up to the lightning discharge.