A　simple　and　efficient　interface-fitted　mesh　generation　algorithm　which　can　produce　a　semi-structured　interface-fitted　mesh　in　two　and　three　dimensions　quickly　is　developed　in　this　paper.　Elements　in　such　interface-fitted　meshes　are　not　restricted　to　simplices　but　can　be　polygons　or　polyhedra.　Especially　in　3D,　the　polyhedra　instead　of　tetrahedra　can　avoid　slivers.　Virtual　element　methods　are　applied　to　solve　elliptic　interface　problems　with　solutions　and　flux　jump　conditions.　Algebraic　multigrid　solvers　are　used　to　solve　the　resulting　linear　algebraic　system.　Numerical　results　are　presented　to　illustrate　the　effectiveness　of　our　method.　(C)　2017　Elsevier　Inc.　All　rights　reserved.