  We consider Riemann sum approximations of stochastic integrals with respect
to the fractional Browian motion of index $H\geq \frac12$. We show the
convergence of these schemes at first and second order. The processes obtained
in the limit in the second case are stochastic integrals with respect to the
Rosenblatt process if $H >\frac34$ and the standard Brownian motion otherwise.
These results are obtained under the assumption that the integrand is a
`controlled' process. We provide many examples of such processes, in particular
fractional semimartingales and multiple Wiener-It\^o integrals
