The superlative adhesive properties of some biological attachment systems, such as those of geckos, spiders, and insects, have inspired researchers from different fields (e.g. biology, physics and engineering) to conceive and design man-made microstructured surfaces that might mimic their performance. Among the several proposed designs, very recently mushroom-shaped adhesive microstructures have drawn the interest of scientists and engineers, because experiments have proved their superiority compared to other micro- and nano-structures. In this article, we explain theoretically the physical mechanism behind the enhanced adhesion of such microstructures, and provide for the first time a useful tool to predict adhesive performance depending on the geometry, mechanical properties of the material, and energy of adhesion. Our theoretical predictions are strongly supported by the available experimental data. The present study can streamline the optimisation of adhesive microstructures for industrial applications.