The aim of this paper is to present results on actions of compact Lie groups on Stein spaces. The main result is the following: Complexification Theorem. Let K be a compact Lie group and \(K^{{\mathbb{C}}}\) a complexification of K. If K acts on a reduced Stein space X, then there exists a complex space \(X^{{\mathbb{C}}}\) with a holomorphic action \(K^{{\mathbb{C}}}\times X^{{\mathbb{C}}}\to X^{{\mathbb{C}}}\) and a K-equivariant holomorphic map i: \(X\to X^{{\mathbb{C}}}\) with the following properties: (i) i: \(X\to X^{{\mathbb{C}}}\) is an open embedding and i(X) is a Runge subset of \(X^{{\mathbb{C}}}\) such that \(K^{{\mathbb{C}}}\cdot i(X)=X^{{\mathbb{C}}}.\) (ii) \(X^{{\mathbb{C}}}\) is a Stein space. (iii) If \(\Phi\) is a K-equivariant holomorphic map from X into a complex space Y on which \(K^{{\mathbb{C}}}\) acts holomorphically, then there exists a unique \(K^{{\mathbb{C}}}\)-equivariant holomorphic map \(\Phi^{{\mathbb{C}}}: X^{{\mathbb{C}}}\to Y\) such that the diagram commutes.