Laminar flow past a sphere rotating in the transverse direction is numerically investigated in order to understand the effect of the rotation on the characteristics of flow over the sphere. Numerical simulations are performed at Re=100, 250 and 300, where the Reynolds number is based on the tree-stream velocity and the sphere diameter. The rotational speeds considered are in the range of $0{leq}{omega}^*{leq}1.2$, where ${omega}^*$ is the maximum velocity on the sphere surface normalized by the tree-stream velocity. Without rotation, the flow past a sphere experiences steady axisymmetry, steady planar-symmetry, and unsteady planar-symmetry, respectively, at Re=100, 250 and 300. With rotation, however, the flow becomes planar-symmetric for all the cases investigated, and the symmetry plane of flow is orthogonal to the rotational direction. Also, the rotation affects the flow unsteadiness, and its effect depends on the rotational speed and the Reynolds number. The flow is steady irrespective of the rotational speed at Re=100, whereas at Re=250 and 300 it undergoes a sequence of transitions between steady and unsteady flows with increasing ${omega}^*$. As a result, the characteristics of vortex shedding and vortical structures in the wake are significantly modified by the rotation at Re=250 and 300. For example, at Re=300, vortex shedding occurs at low values of ${omega}^*$, but it is completely suppressed at ${omega}^*=0.4$ and 0.6. Interestingly, at ${omega}^*=1$ and 1.2, unsteady vortices are newly generated in the wake due to the shear layer instability. The critical rotational speed, at which the shear layer instability begins to occur, is shown to be higher at Re=250 than at Re=300.