A group of PhD students at Columbia University are developing a system called Cider that allows iOS apps to run on Android. It essentially tricks iOS apps into thinking they are running within their native framework. Rather than using the typical emulator or virtual machine set-up you might be imagining, Cider adapts code on the fly so that it will work with the Android kernel and libraries.
As you can see in the video below, the team demonstrates Cider in action, running apps like Yelp and Apple's iBooks on a Nexus 7. Another thing you'll see is that the performance isn't great (yet), in fact it isn't very good at all with the iOS apps running very sluggish. However, this is something to be expected, other examples of this kind of technology from established companies have been less then stellar.
Another issue with Cider at this point comes with the additional hardware features many iOS apps have built-in. The compatibility layer does not yet support hardware integrations like cameras, cell phone radios and Bluetooth yet, but the 6 man team at Columbia said it has since added support for GPS features. Many apps will simply not function corectly, where as others will just continue to work without the particular unsupported feature.
The team, Jeremy Andrus, Alexander Van't Hof, Naser AlDuaij, Christoffer Dall, Nicolas Viennot, and Jason Nieh, plan to continue research and development on Cider. You can see the complete published research paper here.