The hybrid simulations, where the ions are treated kinetically and the electrons as a fluid, seek to describe ion microphysics with maximum physical fidelity. The hybrid approach addresses the fundamental need for space plasma models to incorporate physics beyond magnetohydrodynamics. Global hybrid simulations must account for a wide range of both kinetic ion and whistler/Alfvén wave spatio-temporal scales in strongly inhomogeneous plasmas. We present results from two three-dimensional hybrid simulations performed with a novel asynchronous code, HYPERS designed to overcome computational bottlenecks that typically arise in such multiscale simulations. First, we demonstrate an excellent match between simulated lunar wake profiles and observations. We also compare our simulations with two other simulations performed with conventional (time-stepped) hybrid codes. Second, we investigate the interaction of the solar wind with the Earth's dayside magnetosphere under conditions when the orientation of the interplanetary magnetic field is quasi-radial. In this high-resolution simulation we highlight three-dimensional properties of foreshock perturbations formed by the backstreaming ions.