We study the classical electron motion of two- electron atoms. We present techniques to regularize singularities in the equations of motion and to calculate stability properties of periodic orbits in six-dimensional phase space. Depending on the configuration of the electron pair the classical phase space turns to be regular, purely chaotic, or of mixed structure. For the fully hyperbolic case of collinear electron motion a strong correlation between ionizing trajectories and the collision manifold of the triple collision is found. For the near- integrable collinear motion with both electrons on the same side of the atom we (numerically) derive an action-angle representation of the Hamiltonian.