A mathematical theory of ball lightning is presented. Ball lightning is considered to be an externally powered dc phenomenon that results from the funneling of cloud‐to‐ground current through a region of high temperature and high electrical conductivity. An approximate solution of the equations of current conservation and energy conservation with proper boundary conditions yields a luminous region of air. The predicted ball lightning diameters are in good agreement with observation for reasonable values of current, current density, electric field, and electrical conductivity inside and outside the ball. The luminosity, energy content, formation, locale, motion, and observed modes of disappearance of ball lightning are discussed. An explanation for the phenomenon of bead lightning is advanced.