We present results from a new hybrid-Vlasov simulation code, Vlasiator, designed for global magnetospheric simulations. Vlasiator represents ions by a six-dimensional distribution function propagated using a finite volume approach. The distribution functions are self-consistently coupled to electromagnetic fields with electrons modeled as a charge-neutralizing fluid. A novel sparse representation of the distribution function reduces the computational demands of the problem by up to two orders of magnitude. The capabilities of the code are demonstrated by reproducing characteristics of the ion/ion right-hand resonant beam instability, as well as key features of the collisionless bow shock and magnetosheath in front of the Earth's magnetosphere in global five-dimensional (two in ordinary space, three in velocity space) simulations. We find that Vlasiator reproduces the ion velocity distribution functions with quality comparable to spacecraft observations.