Let \(F\) be a meromorphic function in several complex variables. The authors say that \(F\) has a factorization with left factor \(f\) and right factor \(g\) if \(F(z) = f(g(z))\), \(z \in \mathbb{C}^ n\), where \(f\) is a meromorphic function from \(\mathbb{C}\) to \(\mathbb{P}^ 1\) and \(g\) is an entire function of several complex variables. If every factorization of \(F\) implies that \(f\) is bilinear in \(\mathbb{C}\) or \(g\) is linear in \(\mathbb{C}^ n\) \((f\) is rational or \(g\) is a polynomial), then \(F\) is called prime (pseudo-prime). The authors study when a meromorphic function in several complex variables is prime under composition. In \S2 and \S3 they obtain two theorems on this subject for pseudo-primeness of meromorphic solutions of linear PDE's and primeness for entire functions of finite order with algebraic divisor. In \S4 relationships among the growth of \(f(g)\), \(f\) and \(g\) are studied and the solution of an open problem of \textit{W. D. Brownawell} [Can. J. Math. 39, No. 4, 825-834 (1987; Zbl 0631.35008)] is given.