In this paper, we give a generalization of normal curves to n-dimensional Euclidean space. Then we obtain a necessary and sufficient condition for a curve to be a normal curve in the n-dimensional Euclidean space. We characterize the relationship between the curvatures for any unit speed curve to be congruent to a normal curve in the n-dimensional Euclidean space. Moreover, the differentiable function f ( s) is introduced by using the relationship between the curvatures of any unit speed curve in En. Finally, the differential equation characterizing a normal curve can be solved explicitly to determine when the curve is congruent to a normal curve.