Spontaneously broken gauge theories in a constant external electromagnetic field are shown to exhibit a first-order phase transition to a restored symmetry phase when the external field exceeds a certain critical value. The effects of fields characterized by various values of the two Lorentz invariants F 1 = 1 2 (B 2 − E 2) and F 2 = E · B are discussed. In a simple SU(2) model the critical field strength is found to be g R 2( F 1) crit = 0.057 m w 4, m w being the vector boson mass. A number of theoretical developments in the background field formalism are presented. A new gauge-fixing term, the background field R gauge, is introduced. The configuration space heat kernel method for evaluating functional determinants, extended to allow the use of dimensional regularization, is employed, and it is shown how to perform background field calculations in a gauge specified by an arbitrary parameter α. Further applications of these methods are discussed.