Kinetochores provide chromosomes with points of attachment to spindle microtubules during cell division, and are therefore essential for genome inheritance and the propagation of life. In addition to binding microtubules, kinetochores control mitotic surveillance mechanisms that promote chromosome bi-orientation (the error correction mechanism) and prevent premature mitotic exit in presence of incomplete or incorrect microtubule attachments (spindle assembly checkpoint, SAC). Elimination of the NDC80 complex, the main microtubule receptor of kinetochores, causes a SAC deficiency, identifying this complex as a crucial regulatory focus for checkpoint function. In recent years, there has been considerable progress in understanding how the SAC effector, known as the mitotic checkpoint complex (MCC), assembles from its individual components to inhibit its target, the anaphase promoting complex/cyclosome (APC/C). Conversely, how microtubule attachment to kinetochores regulates the SAC remains incompletely understood. From a molecular perspective, answering this question implies investigating the mechanisms that promote targeting of the SAC proteins to unattached kinetochores, and suppress it upon microtubule binding and biorientation. In our recent work, we have combined biochemical reconstitutions, structural biology/modelling, and cell biology to gain insights into this fundamental biological question.