Z-eigenvalues　of　tensors,　especially　extreme　ones,　are　quite　useful　and　are　related　to　many　problems,　such　as　automatic　control,　quantum　physics,　and　independent　component　analysis.　For　supersymmetric　tensors,　calculating　the　smallest/largest　Z-eigenvalue　is　equivalent　to　solving　a　global　minimization/maximization　problem　of　a　homogenous　polynomial　over　the　unit　sphere.　In　this　paper,　we　utilize　the　sequential　subspace　projection　method　(SSPM)　to　find　extreme　Z-eigenvalues　and　the　corresponding　Z-eigenvectors.　The　main　idea　of　SSPM　is　to　form　a　2-dimensional　subspace　at　the　current　point　and　then　solve　the　original　optimization　problem　in　the　subspace.　SSPM　benefits　from　the　fact　that　the　2-dimensional　subproblem　can　be　solved　by　a　direct　method.　Global　convergence　and　linear　convergence　are　established　for　supersymmetric　tensors　under　certain　assumptions.　Preliminary　numerical　results　over　several　testing　problems　show　that　SSPM　is　very　promising.　Besides,　the　globalization　strategy　of　random　phase　can　be　easily　incorporated　into　SSPM,　which　promotes　the　ability　to　find　extreme　Z-eigenvalues.　Copyright　(c)　2014　John　Wiley　&　Sons,　Ltd.