We demonstrate a general principle that hybrid structures of any sort inevitably will give rise to a pure spin current flowing parallel to the interface region when a charge current is injected. This stems from the broken mirror symmetry near the interface which gives rise to spin-orbit coupling that deflects incoming electrons in a spin-discriminating fashion. We establish a general analytical condition for the appearance of this effect, and calculate the transverse spin current explicitly using two different models. In addition, we investigate how the process of Andreev reflection influences this phenomenon in the scenario where one of the materials is superconducting.