  We discuss the continuity of the composition on several spaces of holomorphic
mappings on open subsets of a complex Banach space. On the Fr\'{e}chet space of
the entire mappings that are bounded on bounded sets the composition turns to
be even holomorphic. In such space we consider linear subspaces closed under
left and right composition. We discuss the relationship of such subspaces with
ideals of operators and give several examples of them. We also provide natural
examples of spaces of holomorphic mappings where the composition is not
continuous.
