Vertical coupling between waveguides is a critical component for three-dimensional (3-D) integrated optics. Vertical integration adds flexibility in integrating different devices that require different materials, and facilitates coupling of the miniature devices with optical fibers. We propose a systematic design of a relatively simple and versatile vertical coupler that provides not only vertical interconnection, but also mode-size transformation and polarization mode selection all in one. As a vertical polarization splitter, it separates the TE and TM polarizations onto different vertical levels of a 3-D photonics structure, and is thus uniquely different from conventional splitters based on directional couplers or other planar devices. The vertical coupler consists of a larger bottom waveguide that serves to improve the fiber coupling, and a smaller top waveguide that contains the actual photonic device. As a polarization-independent coupler the vertical coupler is shown to transfer light with more than 90% efficiency for all polarizations over a transfer length of only 150 µm. As a polarization mode splitter, the vertical coupler preferentially couples TE or TM polarization with a contrast ratio up to 20dB. This versatility renders the vertical coupler a compact and useful input-stage device that improves the fiber coupling to small active devices and also provides a mechanism of polarization control.