The evolute of a given curve is a well-known concept in 3-dimensional Euclidean space |R3. In this paper, we generalize this concept to n-dimensional Euclidean space |Rn . In addition, defining an involute-evolute curve couple (a, (3) in |Rn , we obtaiıı some properties connected with (a, j3) and its horizontal lift (a ^ , |3H ) in T|Rn .