The experiments described in this review show that, when the 3He superfluids are rotated, many different vortex structures can be created. The vortex cores are large enough so that several details in their structure can be observed experimentally. When probed with NMR, the vortices in 3He-B display both spontaneous and induced magnetization; their large magnetic moments reveal the existence of non-trivial vortex cores, consisting of magnetic superfluids. A first order phase change in the core structure, along a well defined transition line on the P-T-plane, appears a discontinuity in the magnetization of the rotating 3He-B. A similar transition, marked by a jump in the critical velocity for vortex creation, is also observed in hydrodynamic experiments. The most easily created vortices in 3He-A are doubly quantized and continuous. This identification is compatible with the results of NMR and negative ion experiments. The latter work reveals a focusing interaction between the ions and the vortex textures. Another, presumably singular vortex was also found in 3He-A by the ion technique in high magnetic fields. Gyroscopic measurements have demonstrated persistent currents in both superfluid phases of 3He. In addition, experimental data on vortex-free counterflow states, on critical velocities, on vortex states of 3He-A in restricted geometries and on mutual friction between the vortex lattice and the normal fluid are discussed.