To maximize the matching between a coaxial cable and rectangular waveguides, we present a computational topology optimisation approach that decides for each point in a given domain whether to hold a good conductor or a good dielectric. The conductivity is determined by a gradient-based optimisation method that relies on finite-difference time-domain solutions to the 3D Maxwell's equations. Unlike previously reported results in the literature for this kind of problems, our design algorithm can efficiently handle tens of thousands of design variables that can allow novel conceptual waveguide designs. We demonstrate the effectiveness of the approach by presenting optimised transitions with reflection coefficients lower than -15 dB over more than a 60% bandwidth, both for right-angle and end-launcher configurations. The performance of the proposed transitions is cross-verified with a commercial software, and one design case is validated experimentally.