In this paper we consider the norm of composition operators on vector-valued function spaces. Let X be a complex Banach space and \(\theta \) be an inner function. We show that the norm of the composition operator \(C_{\theta }\) on the vector-valued Hardy space \(H^p(\mathbb T, X)\) ( \(1 \le p < \infty \) ) is given by \(\begin{aligned} ||C_{\theta }||_{{\mathcal {B}}(H^p(\mathbb T,\,X))}=\left( \frac{1+|\theta (0)|}{1-|\theta (0)|}\right) ^{\frac{1}{p}}. \end{aligned}\) Also we are concerned with the Poisson integral of vector-valued function on the unit circle \(\mathbb T\) and showed some properties in composition operators on vector-valued function spaces, and operator-valued function spaces.
