We　establish　the　stability/instability　criteria　for　the　stratified　compressible　magnetic　Rayleigh–Taylor　(RT)　problem　in　Lagrangian　coordinates.　More　precisely,　under　the　stability　condition　Ξ＜　1　(under　such　case,　the　third　component　of　impressed　magnetic　field　is　not　zero),　we　show　the　existence　of　a　unique　solution　with　an　algebraic　decay　in　time　for　the　(compressible)　magnetic　RT　problem　with　proper　initial　data.　The　stability　result　in　particular　shows　that　a　sufficiently　large　(impressed)　vertical　magnetic　field　can　inhibit　the　growth　of　the　RT　instability.　On　the　other　hand,　if　Ξ＞　1　,　there　exists　an　unstable　solution　to　the　magnetic　RT　problem　in　the　Hadamard　sense.　This　in　particular　shows　that　the　RT　instability　still　occurs　when　the　strength　of　an　magnetic　field　is　small　or　the　magnetic　field　is　horizontal.　Moreover,　by　analyzing　the　stability　condition　in　the　magnetic　RT　problem　for　vertical　magnetic　fields,　we　can　observe　that　the　compressibility　destroys　the　stabilizing　effect　of　magnetic　fields　in　the　vertical　direction.　Fortunately,　the　instability　in　the　vertical　direction　can　be　inhibited　by　the　stabilizing　effect　of　the　pressure,　which　also　plays　an　important　role　in　the　proof　of　the　stability　of　the　magnetic　RT　problem.　In　addition,　we　extend　the　results　for　the　magnetic　RT　problem　to　the　compressible　viscoelastic　RT　problem,　and　find　that　the　stabilizing　effect　of　elasticity　is　stronger　than　that　of　the　magnetic　fields.　©　2019,　Springer-Verlag　GmbH　Germany,　part　of　Springer　Nature.