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Prior to cell division, chromosomes are segregated by a precise molecular machine of the mitotic spindle. Spindle self-assembles remarkably fast in an elegant process of search and capture, during which dynamic microtubules grow and shrink rapidly and repeatedly until a contact with kinetochores – specialized organelles on the chromosomes is established. Computer simulations demonstrate that this random search and capture process is not fast enough; more importantly, it leads to an enormous number of microtubule-kinetochore attachment errors leading to cancer and death. Recent simulations and experiments indicate that the search is accelerated and its errors are corrected and prevented by a number of redundant mechanisms including rapid dissolution of wrong attachments, forced positioning of the chromosomes in optimal configurations and chromosome rotations. Furthermore, self-assembly mechanisms are universal, which will be demonstrated on the example of Golgi apparatus formation.