  We study biharmonic maps and f-biharmonic maps from a round sphere $(S^2,
g_0)$, the latter maps are equivalent to biharmonic maps from Riemann spheres
$(S^2, f^{-1}g_0)$. We proved that for rotationally symmetric maps between
rotationally symmetric spaces, both biharmonicity and f-biharmonicity reduce to
a 2nd order linear ordinary differential equation. As applications, we give a
method to produce biharmonic maps and f-biharmonic maps from given biharmonic
maps and we construct many examples of biharmonic and f-biharmonic maps from a
round sphere $S^2$ and between two round spheres. Our examples include
non-conformal proper biharmonic maps $(S^2, f^{-1}g_0)\longrightarrow S^2$ and
$(S^2, f^{-1}g_0)\longrightarrow S^n$, or non-conformal f-biharmonic maps
$(S^2, g_0)\longrightarrow S^2$ and $(S^2,g_0)\longrightarrow S^n$ from round
sphere with two singular points.
