Structural　reliability　analysis　involving　highly　nonnormally　distributed　performance　functions　or　relatively　small　failure　probabilities　is　still　a　challenging　problem　in　engineering　practice.　To　overcome　these　problems,　this　paper　developed　the　moment　method　combined　with　Box-Cox　transformation.　A　new　criterion　is　proposed　to　select　the　optimal　Box-Cox　transformation　parameter　that　makes　the　transformed　performance　functions　approximately　normally　distributed.　This　criterion　was　established　based　on　the　minimum　distance　between　the　pair　of　the　skewness　and　kurtosis　of　the　transformed　performance　function　and　that　of　a　normal　random　variable,　in　which　the　statistical　moments　of　the　transformed　performance　function　are　evaluated　from　the　point-estimate　method.　The　failure　probability　was　evaluated　using　a　third-order　moment　reliability　index.　The　accuracy　and　efficiency　of　the　proposed　method　were　demonstrated　through　several　numerical　examples,　including　structural　reliability　analysis　with　correlated　random　variables;　stochastic　processes;　and　nonnormal,　nonlinear,　and　implicit　performance　functions.