Evaluating　the　failure　probabilities　with　rare　events　remains　a　challenging　task,　especially　when　dealing　with　complex　limit　state　functions　with　high-dimensional　random　inputs.　To　handle　this　problem,　an　adaptive　polynomial　skewed-normal　transformation　(A-PSNT)　model　is　proposed　in　this　paper　for　evaluating　the　probability　distribution　of　the　limit　state　function　and　failure　probability.　First,　the　polynomial　transformation　models　on　the　basis　of　normal　distribution　are　introduced,　where　the　limitations　are　also　pointed　out.　Then,　the　skewed-normal　distribution　is　presented,　which　serves　as　the　basis　of　the　proposed　polynomial　transformation　model.　The　PSNT　with　a　given　order　can　be　initiated,　whereby　the　unknown　coefficients　can　be　determined　by　using　the　probability　weighted　moments　(PWMs)-matching　technique.　Further,　a　tail　error　criterion　is　proposed　to　select　the　most　appropriate　order　for　the　PSNT　model,　which　makes　the　proposed　method　an　adaptive　one.　To　handle　the　low-to　high-dimensional　random　inputs,　two　low-discrepancy　sampling　methods　are　employed　to　generate　samples　for　unbiasedly　estimating　the　PWMs　of　the　limit　state　function.　Once　the　probability　distribution　of　the　limit　state　function　is　reconstructed　by　the　proposed　A-PSNT　model,　the　failure　probability,　especially　for　the　rare　one,　can　be　determined　accordingly.　The　proposed　method　is　validated　through　numerical　examples　involving　both　static　and　dynamic,　low-to　high-dimensional　problems,　where　some　classical　reliability　analysis　methods　are　also　adopted　for　comparisons.　The　results　show　that　the　proposed　A-PSNT　model　not　only　can　effectively　reconstruct　the　probability　distribution,　but　also　is　able　to　accurately　predict　the　failure　probability　with　rare　events.