The　global　well-posedness　of　a　new　class　of　initial-boundary　value　problem　on　incompressible　MHD　equations　in　the　bounded　domain　with　the　smooth　boundary　is　studied.　The　existence　of　a　class　of　global　weak　solution　to　the　initial　boundary　value　problem　for　two/three-dimensional　incompressible　MHD　equation　with　the　given　pressure-velocity＇s　relation　boundary　condition　for　the　fluid　field　at　the　boundary　and　with　one　perfectly　insulating　boundary　condition　for　the　magnetic　field　at　the　boundary　is　obtained,　and　the　global　existence　and　uniqueness　of　the　smooth　solution　to　the　corresponding　problem　in　two-dimensional　case　for　the　smooth　initial　data　is　also　established.　The　corresponding　results　are　also　extended　to　the　two/three-dimensional　incompressible　MHD-Boussinesq　equations　with　the　density-velocity＇s　relation　boundary　condition　for　the　density.　©　2023　Elsevier　Inc.