This chapter describes the construction of new solutions V of the Korteweg-deVries (KdV) hierarchy of equations by deformations of a given finite-gap solution V0. To describe the nature of these deformations the chapter assumes a moment that the given real-valued quasi-periodic finite-gap solution V0 is described in terms of the Its–Matveev formula. The chapter also gives brief account of the KdV hierarchy using a recursive approach ans describes real-valued quasi-periodic finite-gap solutions and the underlying Its–Matveev formula in some detail. To describe the hierarchy of KdV equations the chapter first recalls the recursive approach to the underlying Lax pairs. In addition, the chapter introduces isospectral and non-isospectral deformations in a systematic way by alluding to single and double commutation techniques. © 1993, Academic Press Inc.