In this paper we study the motion of curves in $3$-dimensional spherical space $\mathbb{S}^{3}$. We derive the evolution equations of the orthonormal frame and evolution equations for curvatures. Moreover, we give some explicit examples of motions of inextensible curves in $\mathbb{S}^{3}$ and we determine the curves from their intrinsic equations (curvature and torsion). Then we determine the surfaces that are generated by the motion of these curves. To visualize these surfaces in $\mathbb{S}^{3}$, we use the stereographic projection in $\mathbb{R}^{4}$.