Summary: This article is the first of two concerned with the development of the theory of equations of KdV type from the point of view of twistor theory and the self-dual Yang-Mills equations. A hierarchy on the self-dual Yang-Mills equations is introduced and it is shown that a certain reduction of this hierarchy is equivalent to the \(n\)-generalized KdV- hierarchy. It also emerges that each flow of the \(n\)-KdV hierarchy is a reduction of the self-dual Yang-Mills equations with gauge group \(\text{SL}_ n\). It is further shown that solutions of the self-dual Yang-Mills hierarchy and their reductions arise via a generalized Ward transform from holomorphic vector bundles over a twistor space. Explicit examples of such bundles are given and the Ward transform is implemented to yield a large class of explicit solutions of the \(n\)-KdV equations. It is also shown that the construction of Segal and Wilson of solutions of the \(n\)-KdV equations from loop groups is contained in our approach as an ansatz for the construction of a class of holomorphic bundles on twistor space. In the second part the twistor description developed here will be used to investigate other aspects of the theory of equations of KdV type.