This paper concerns the problem proposed by J.-P. Serre whether a holomorphic fiber bundle with Stein fiber and Stein base is itself Stein. Let \(G\) be a real connected Lie group acting on a complex manifold \(X\) as holomorphic diffeomorphisms. The authors first define a complexification \(G^\mathbb{C}\) of \(G\) and then a globalization \(X^*\) of the induced local \(G^\mathbb{C}\)-action on \(X\) for the universal complexification \(G^\mathbb{C}\) of \(G\). Next, a universal globalization is defined. Assume that there exists a cocompact discrete subgroup \(\Gamma\) of \(G\) such that \(G^\mathbb{C}/ \Gamma\) is a Stein manifold. The authors prove that the following three statements (1)--(3) are equivalent: (1) the universal globalization \(X^*\) is a Stein, (2) \((G^\mathbb{C} \times X)/ \Gamma\) is Stein, (3) if \(E(X,\pi,B,G)\) is a holomorphic fiber bundle with Stein base \(B\), Stein fiber \(X\), and a structure group \(G\), then \(E\) is Stein. As a corollary the authors asser that if \(E(X, \pi,B,K)\) is a holomorphic fiber bundle with Stein base \(B\), Stein fiber \(X\) and compact structure group \(K\) then \(E\) is Stein.