The spins, moments and radii of krypton isotopes have been investigated by collinear fast-beam laser spectroscopy in combination with ultra-sensitive collisional ionization detection. The sequence of isotopes under study ranges from the neutron-deficient N = Z = 36 isotope 72Kr to the neutron-rich 96Kr (N = 60). The mean-square charge radii in the neighbourhood of the N = 50 neutron-shell closure exhibit a pronounced shell effect which has recently been explained in the framework of relativistic mean-field theory. The results for the neutron-deficient nuclei are related to the shape coexistence of strongly prolate and near-spherical states which is known from nuclear spectroscopy. Here, an inversion of the odd-even staggering is observed below the neutron number N = 45. The neutron-rich transitional nuclei are influenced by the N = 56 subshell closure. In contrast to the N = 60 isotones 97Rb, 98Sr and 100Zr, the new isotope 96Kr is not strongly deformed.