Recent experiments have demonstrated that radiative heat transfer between objects separated by nanometre-scale gaps considerably exceeds the predictions of far-field radiation theories. Exploiting this near-field enhancement is of great interest for emerging technologies such as near-field thermophotovoltaics and nano-lithography because of the expected increases in efficiency, power conversion or resolution in these applications. Past measurements, however, were performed using tip-plate or sphere-plate configurations and failed to realize the orders of magnitude increases in radiative heat currents predicted from near-field radiative heat transfer theory. Here, we report 100- to 1,000-fold enhancements (at room temperature) in the radiative conductance between parallel-planar surfaces at gap sizes below 100 nm, in agreement with the predictions of near-field theories. Our measurements were performed in vacuum gaps between prototypical materials (SiO2-SiO2, Au-Au, SiO2-Au and Au-Si) using two microdevices and a custom-built nanopositioning platform, which allows precise control over a broad range of gap sizes (from <100 nm to 10 μm). Our experimental set-up will enable systematic studies of a variety of near-field-based thermal phenomena, with important implications for thermophotovoltaic applications, that have been predicted but have defied experimental verification.