This is an international, multicenter, open-label, long-term safety study of ZX008 in patients with epileptic encephalopathy, including Dravet syndrome or Lennox-Gastaut syndrome. Subjects eligible for participation are those with Dravet syndrome who are currently enrolled in Study ZX008-1503, or those with Lennox-Gastaut syndrome who have successfully completed Study ZX008-1601-Part 2, and are candidates for continued treatment with ZX008 for an extended period of time, or those with Dravet syndrome, Lennox-Gastaut syndrome, or another epileptic encephalopathy who have completed participation in another Zogenix-sponsored study and have been invited to participate in this study.