In this paper we distinguish between free Hessenberg forms, where only the system matrix is transformed into a special structure, and system Hessenberg forms, where either an input or output matrix is simultaneously simplified. We show how the freedom in the choice of the transformation matrix for the free Hessenberg form can be used to obtain a unique system Hessenberg form. We discuss numerical aspects and outline some basic applications for both types of Hessenberg forms in control and system theory. We show that the system Hessenberg form can be extremely sensitive to perturbations, a property not necessarily shared by the free Hessenberg form. Several examples are explored, and a first-order perturbation analysis is given. Analogous aspects are also discussed for the Hessenberg/triangular forms resulting from reduction of a matrix pencil. The main emphasis of this paper is on systems with one input.