The cell interior is in constant movement, which is to a large extent determined by microtubules, thin and long filaments that permeate the cytoplasm. To move large objects, microtubules need to connect them to the site of their destination. For example, during cell division, microtubules connect chromosomes with the spindle poles via kinetochores, protein complexes on the chromosomes. A general question is how microtubules, while being bound to one structure, find the target that needs to be connected to this structure. Here we review the mechanisms of how microtubules search for kinetochores, with emphasis on the recently discovered microtubule feature to explore space by pivoting around the spindle pole. In addition to accelerating the search for kinetochores, pivoting helps the microtubules to search for cortical anchors, as well as to self-organize into parallel arrays and asters to target specific regions of the cell. Thus, microtubule pivoting constitutes a mechanism by which they locate targets in different cellular contexts.