We consider a stochastic functional differential equation with infinite memory driven by a fractional Brownian motion with Hurst parameter $H>1/2$. We prove an existence and uniqueness result of the solution to the stochastic differential equation. We investigate the dependence of the solution on the initial condition and the existence of finite moments of the solution. Furthermore we generalize these results to wider classes of stochastic differential equations. The stochastic integral with respect to fractional Brownian motion is defined as a pathwise Riemann-Stieltjes integral.