The crystal and magnetic structures of the spinel compounds ZnCr2 S4 and ZnCr2 Se4 were investigated by high-resolution powder synchrotron and neutron diffraction. ZnCr2 Se4 exhibits a first-order phase transition at TN =21 K into an incommensurate helical magnetic structure. Magnetic fluctuations above TN are coupled to the crystal lattice as manifested by negative thermal expansion. Both the complex magnetic structure and the anomalous structural behavior can be related to magnetic frustration. Application of an external magnetic field shifts the ordering temperature and the regime of negative thermal expansion toward lower temperatures. Thereby, the spin ordering changes into a conical structure. ZnCr2 S4 shows two magnetic transitions at TN1 =15 K and TN2 =8 K that are accompanied by structural phase transitions. The crystal structure transforms from the cubic spinel-like (space group Fd 3̄ m) at high temperatures in the paramagnetic state, via a tetragonally distorted intermediate phase (space group I 41 /amd) for TN2 <T< TN1 into a low-temperature orthorhombic phase (space group Imma) for T< TN2. The cooperative displacement of sulfur ions by exchange striction is the origin of these structural phase transitions. The low-temperature structure of ZnCr2 S4 is identical to the orthorhombic structure of magnetite below the Verwey transition. When applying a magnetic field of 5 T the system shows an induced negative thermal expansion in the intermediate magnetic phase as observed in ZnCr2 Se4.