This　work　is　a　continuation　of　our　previous　work　(Shao,　Nonlinear　Anal.　Real　World　Appl.　11　(2010)　3791-3808)　＇Global　structure　stability　of　Riemann　solutions　for　linearly　degenerate　hyperbolic　conservation　laws　under　small　BV　perturbations　of　the　initial　data＇.　In　the　present　paper,　we　prove　the　global　structure　instability　of　the　Lax＇s　Riemann　solution　u=U(x/t)　containing　only　shocks　and　contact　discontinuities,　and　at　least　a　shock　wave,　of　general　n×n　quasilinear　hyperbolic　systems　of　conservation　laws　under　small　BV　perturbations　of　the　Riemann　initial　data.　The　perturbations　are　in　BV　but　they　are　assumed　to　be　C1　smooth,　with　bounded　and　possibly　large　C　1　norms.　We　prove　the　non-existence　of　global　piecewise　C1　solution　to　a　class　of　the　generalized　Riemann　problem,　which　can　be　regarded　as　a　small　BV　perturbation　of　the　corresponding　Riemann　problem,　for　general　n×n　quasilinear　hyperbolic　systems　of　conservation　laws.　Some　applications　to　quasilinear　hyperbolic　systems　of　conservation　laws　arising　in　physics,　particularly　to　one-dimensional　Euler　equations　of　gas　dynamics　for　a　compressible,　inviscid,　non-heat　conducting　gas　in　Eulerian　coordinates,　are　also　given.　©　2012　The　Author　2012.　Published　by　Oxford　University　Press　on　behalf　of　the　Institute　of　Mathematics　and　its　Applications.　All　rights　reserved.