In　this　paper,　boundary　and　interior　crises　in　a　fractional-order　piecewise　system　are　studied　using　the　extended　generalized　cell　mapping　(EGCM)　method　as　a　system　control　parameter　varies.　A　new　development　of　the　EGCM　method　is　presented　to　deal　with　the　non-smooth　characteristics　of　a　fractional-order　piecewise　system.　A　boundary　crisis　occurs　when　a　chaotic　attractor　collides　with　a　regular　saddle　on　the　basin　boundary　leaving　behind　a　chaotic　saddle　in　the　place　of　the　original　attractor　and　saddle.　With　an　increase　of　the　system　parameter,　both　the　chaotic　saddle　and　the　chaotic　attractor　become　larger　and　finally　touch　each　other　in　the　basin　of　attraction,　which　causes　the　size　change　of　the　chaotic　attractor　suddenly,　namely,　an　interior　crisis　occurs.　Additionally,　the　routes　to　chaos　and　out　of　chaos　for　the　system　are　explored　by　the　EGCM　method.　It　is　found　that　the　route　to　chaos　is　a　period-doubling　bifurcation,　and　out　of　chaos　is　a　saddle-node　bifurcation.　These　results　further　explain　the　dynamics　evolution　of　the　fractional-order　piecewise　system　from　a　global　perspective.　Based　on　this,　it　can　be　demonstrated　that　the　EGCM　method　is　also　effective　for　the　global　dynamics　of　fractional-order　piecewise　systems.　©　2021,　The　Author(s),　under　exclusive　licence　to　Springer　Nature　B.V.　part　of　Springer　Nature.