Transportation networks play a crucial role in human mobility, the exchange of goods, and the spread of invasive species. With 90% of world trade carried by sea, global shipping provides one of the most important modes of transportation. Shipping also constitutes the world largest transportation vector for marine bioinvasion, transferring accidentally numerous species around the world. Here we use information about the itineraries of 16,363 cargo ships during the year 2007 to construct a network of shipping connections between ports. We perform a statistical analysis of the network topology, and we show that the network possesses a heavy-tailed distribution for the connectivity of ports with systematic differences between ship types. Our analysis improves current assumptions based on gravity models of ship movements, an important step towards understanding global patterns of shipping mobility and trade. Coupling the shipping network with biogeography and environmental conditions at the ports, we quantify the likelihood of invasion by the exchange of ballast water. Our model allows to identify high risk invasion routes, hot spots of bioinvasion, and major source regions from which bioinvasion is likely to occur. Our predictions agree with observations in the field at various locations in the world. Finally, motivated from the invasion process, we present a conceptional model for the spread of a binary variable (here: invaded or non-invaded port) on a complex network. Despite its simplicity, the model exhibits complex dynamics and shows many properties that set it apart from similar models of epidemic spread or cascading failures on complex networks.