Zika virus (ZIKV) causes congenital microcephaly. Although ZIKV can impair cell cycle progression and provoke apoptosis, which probably contributes to disease aetiology through depletion of neural progenitor cells, additional cellular mechanisms may be important. Here, we investigated whether ZIKV infection alters centrosome number and spindle positioning, because such defects are thought to be at the root of inherited primary autosomal recessive microcephaly (MCPH). In addition to HeLa cells, in which centrosome number and spindle positioning can be well monitored, we analysed retinal epithelial cells (RPE-1), as well as brain-derived microglial (CHME-5) and neural progenitor (ReN) cells, using immunofluorescence. We established that ZIKV infection leads to supernumerary foci containing centriolar proteins that in some cases drive multipolar spindle assembly, as well as spindle positioning defects in HeLa, RPE-1 and CHME-5 cells, but not in ReN cells. We uncovered similar phenotypes in HeLa cells upon infection with dengue virus (DENV-2), another flavivirus that does not target brain cells and does not cause microcephaly. We conclude that infection with Flaviviridae can increase centrosome numbers and impair spindle positioning, thus potentially contributing to microcephaly in the case of Zika.