This paper contains an extensive survey of results of biharmonic submanifolds in space forms as well as a construction to manufacture new biharmonic but non-harmonic examples in spheres. Such maps are critical points of the bienergy functional, the integral of the norm square of the tension field. The assumption on constancy of the curvature on the range results in explicit formulas for the associated Euler-Lagrange equations. The new examples are products of spheres contained in a larger sphere and the condition of biharmonicity is expressed in terms of the radii of the participating small spheres.