In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized spherical curves in Euclidean (n + 1) space En+1. Further, we introduce some kind of generalized spherical surfaces in Euclidean spaces E-3 and E-4 respectively. We have shown that the generalized spherical surfaces of first kind in E-4 are known as rotational surfaces, and the second kind generalized spherical surfaces are known as meridian surfaces in I. We have also calculated the Gaussian, normal and mean curvatures of these kind of surfaces. Finally, we give some examples.