Van der Waals heterostructures designed by assembling isolated two-dimensional (2D) crystals have emerged as a new class of artificial materials with interesting and unusual physical properties. Here, the multilayer MoS2-WS2 heterostructures with different configurations are reported and their optoelectronic properties are studied. It is shown that the new heterostructured material possesses new functionalities and superior electrical and optoelectronic properties that far exceed the one for their constituents, MoS2 or WS2. The vertical transistor exhibits a novel rectifying and bipolar behavior, and can also act as photovoltaic cell and self-driven photodetector with photo-switching ratio exceeding 10(3). The planar device also exhibits high field-effect ON/OFF ratio (>10(5)), high electron mobility of 65 cm(2)/Vs, and high photoresponsivity of 1.42 A/W compared to that in isolated multilayer MoS2 or WS2 nanoflake transistors. The results suggest that formation of MoS2-WS2 heterostructures could significantly enhance the performance of optoelectronic devices, thus open up possibilities for future nanoelectronic, photovoltaic, and optoelectronic applications.