The aim of this study is to give spinor representation of successive curves in three-dimensional Euclidean space. In three dimensional Euclidean Space, the spinor representations of a curve with unit speed and a successor curve with the same arc length parameter as this curve has been studied. For this, first of all, the curve with unit speed and its successor curve have been corresponded to two different spinors. Then, using the relationships between these curves, the relationships between the spinors corresponding to these curves have been given. Therefore, geometric interpretations of these curves and corresponding spinors have been made. In addition, different spinor equations of the mates and derivatives of spinors have been examined and geometric interpretations of these spinor equations have been given. Then, spinor equations have been obtained in case the successive curves are helices. Consequently, two examples have been given.