Enduring chromosome segregation errors represent potential threats to genomic stability due to eventual chromosome copy number alterations (aneuploidy) and formation of micronuclei—key intermediates of a rapid mutational process known as chromothripsis that is found in cancer and congenital disorders. The spindle assembly checkpoint (SAC) has been viewed as the sole surveillance mechanism that prevents chromosome segregation errors during mitosis and meiosis. However, different types of chromosome segregation errors stemming from incorrect kinetochore–microtubule attachments satisfy the SAC and are more frequent than previously anticipated. Remarkably, recent works have unveiled that most of these errors are corrected during anaphase and only rarely result in aneuploidy or formation of micronuclei. Here, we discuss recent progress in our understanding of the origin and fate of chromosome segregation errors that satisfy the SAC and shed light on the surveillance, correction, and clearance mechanisms that prevent their transmission, to preserve genomic stability.