Most research on water waves has been based on the assumption of irrotational flow, but vorticity can for example be useful for modelling interactions of surface waves with non-uniform currents. While a lot of progress has been made on the 2D problem with vorticity in the last decades, considerably less is known in 3D. In my talk I will give an overview of some recent existence and non-existence results, focusing on doubly periodic waves, that is waves which are periodic in two distinct horizontal directions.