Duality is an exact symmetry of string perturbation theory

Both $\beta$ duality and $R$ duality in the toroidal compactification of heterotic strings are shown to be symmetries of the contribution of arbitrary genus to the free energy. This symmetry relates physics at radius $R$ and coupling $\kappa$ with physics at radius $\frac{{\alpha }^{\prime }}{R}$ and coupling $\kappa {\left(\frac{{\alpha }^{\prime }}{R^2}\right)}^{\frac{d}{2}}$ where $d$ is the number of dimensions compactified. Some comments on the possible breaking of duality by nonperturbative effects are also included.