The aim of this paper is to start a systematic investigation of the existence of angular limits and angular derivatives of holomorphic maps of infinite dimensional Siegel domains in $J^{*}$-algebras. Since $J^{*}$-algebras are natural generalizations of $C^{*}$-algebras, $B^{*}$-algebras, $J{C}^{*}$-algebras, ternary algebras and complex Hilbert spaces, various significant results follow. Examples are given.