Structural　reliability　analysis　involving　the　non-normal　random　variables　are　commonly　encountered　in　practical　engineering.　Furthermore,　some　random　variables　are　often　unknown　probability　distributions,　and　the　probabilistic　characteristic　of　these　variables　may　be　expressed　using　only　statistical　moments.　In　this　contribution,　an　efficient　algorithm　for　structural　reliability　analysis　considering　the　random　variables　with　non-normal　and　unknown　probability　distributions　is　proposed,　in　which　high-order　unscented　transformation　(HUT)　and　fourth-moment　methods　are　adopted　to　efficiently　evaluate　the　structural　reliability　index.　By　introducing　the　Rosenblatt　transformation　and　third-order　polynomial　transformation　(TPT)　techniques,　random　variables　with　non-normal　or　unknown　distributions　are　transformed　into　standard　normal　distributions.　Accordingly,　the　performance　functions　can　be　reconstructed　using　standard　normal　random　variables,　and　HUT　is　then　employed　to　efficiently　estimate　the　first　four　moments　(i.e.,　mean,　standard　deviation,　skewness,　and　kurtosis)　of　the　performance　function.　Based　on　the　Hermite　polynomial　model,　the　complete　expression　of　fourth-moment　reliability　index　is　derived　for　estimating　the　reliability　index　using　the　first　four　moments　of　the　performance　function.　The　efficiency　and　accuracy　of　the　proposed　algorithm　are　demonstrated　through　six　numerical　examples.　The　results　show　that　the　proposed　algorithm　is　applicable　to　estimating　reliability　index　of　both　static　and　dynamic　reliability　problems　involving　explicit　and　implicit　performance　functions.　©　2022　John　Wiley　&　Sons　Ltd.