In　this　paper,　an　ω-convergence　theory　of　filters　in　an　Lω-space　is　established.　By　means　of　the　ω-convergence　theory,　some　important　characterizations　with　respective　to　the　ω-closed　sets,　ωT2　separation　and　(ω1,　ω2)-continuous　mappings　are　obtained.　Moreover,　the　mutual　relationships　among　ω-convergence　of　molecular　nets,　ω-convergence　of　ideals　and　ω-convergence　of　filters　are　given.　©　2007　Springer-Verlag　Berlin　Heidelberg.