Riemann-Roch theorems assert that certain algebraically defined wrong way maps (transfers) in algebraic K--theory agree with topologically defined ones (see Baum, Fulton, MacPherson, Acta Math. 1979). Bismut and Lott (J. Amer. Math. Soc. 1995) proved such a Riemann--Roch theorem where the wrong way maps are induced by the projection of a smooth fiber bundle, and the topologically defined transfer map is the Becker--Gottlieb transfer. We generalize and refine their theorem, and prove a converse stating that the Riemann--Roch condition is equivalent to the existence of a fiberwise smooth structure. In the process, we prove a family index theorem where the K--theory used is algebraic K--theory, and the fiber bundles have topological (not necessarily smooth) manifolds as fibers.