In　this　paper,　an　omega-convergence　theory　of　filters　in　an　L　omega-space　is　established.　By　means　of　the　omega-convergence　theory,　some　important　characterizations　with　respective　to　the　omega-closed　sets,　omega　T　(2)　separation　and　(omega　(1),　omega　(2))-continuous　mappings　are　obtained.　Moreover,　the　mutual　relationships　among　omega-convergence　of　molecular　nets,　omega-convergence　of　ideals　and　omega-convergence　of　filters　are　given.