We consider electron dynamics in strong electromagnetic fields, such as those expected from the next generation of high-intensity laser facilities. Beginning with a review of constant classical fields, we demonstrate that the electron motion (as given by the Lorentz force equation) can be divided into one of four Lorentz invariant cases. Parameterising the field tensor in terms of a null tetrad, we calculate the radiative energy spectrum for an electron in crossed fields. Progressing to an infinite plane wave, we demonstrate how the electron orbit in the average rest frame changes from figure-of-eight to circular as the polarisation changes from linear to circular. To move beyond a plane wave one must resort to numerics. We therefore present a novel numerical formulation for solving the Lorentz equation. Our scheme is manifestly covariant and valid for arbitrary electromagnetic field configurations. Finally, we reconsider the case of an infinite plane wave from a strong field QED perspective. At high intensities we predict a substantial redshift of the usual kinematic Compton edge of the photon emission spectrum, caused by the large, intensity dependent effective mass of the electrons inside the laser beam. In addition, we find that the notion of a centre-of-mass frame for a given harmonic becomes intensity dependent.