We construct the Nahm transform for Higgs bundles over a Riemann surface of genus at least 2 as hyperholomorphic connections on the total space of the tangent bundle of its dual Jacobian. Roughly speaking, the Nahm transform is a nonlinear analogue of the Fourier transform, transforming anti-self-dual connections on the Euclidean $\mathbb{R}^{4}$ which are invariant under a lattice $\Lambda\subset\mathbb{R}^{4}$ into anti-self-dual connections on the dual space $(\mathbb{R}^{4})^{\vee}$ which are invariant under the dual lattice $\Lambda^{\vee}$. Although the construction is in principle well understood for any subgroup of translations $\Lambda$, analytical details vary for each $\Lambda$. We generalize the Nahm transform of solutions of Hitchin's equations on a 2-dimensional torus $T^{2}$ to solutions of Hitchin's equations on a Riemann surface $\Sigma$ of genus $g\ge2$. More precisely, denoting by $J^{\vee}$ the dual to the Jacobian of $\Sigma$ we prove that the Nahm transform of an irreducible solution of rank at least 2 of Hitchin's equations over a Riemann surface $\Sigma$ of genus $g\ge2$ is a Hermitian vector bundle $\widehat{\mathcal{E}}\to J^{\vee}\times H^{0}(K_{\Sigma})$ equipped with a hyperholomorphic unitary connection $\widehat\nabla$ without flat factors. Moreover, the holomorphic structure induced by $\widehat\nabla$ on $\widehat{\mathcal{E}}$ with respect to a product complex structure on $J^{\vee}\times H^{0}(K_{\Sigma})$ extends to a holomorphic bundle over $J^{\vee}\times\mathbb{P}(H^{0}(K_{\Sigma})\oplus\mathbb{C})$. Closely related to this result is Bonsdorff's Fourier-Mukai transform for stable Higgs bundles on $\Sigma$; in fact our construction is the differential geometric analogue of theis algebraic geometric construction, with the advantage of providing one additional piece of information: the existence of the hyperholomorphic connection without flat factors on $J^{\vee}\times H^{0}(K_{\Sigma})$. This connection and its curvature are explicitly computed and analysed. This is joint work with M. Jardim.