  We perform numerical evolutions of the fully non-linear Einstein-(complex,
massive)Klein-Gordon and Einstein-(complex)Proca systems, to assess the
formation and stability of spinning bosonic stars. In the scalar/vector case
these are known as boson/Proca stars. Firstly, we consider the formation
scenario. Starting with constraint-obeying initial data, describing a dilute,
axisymmetric cloud of spinning scalar/Proca field, gravitational collapse
towards a spinning star occurs, via gravitational cooling. In the scalar case
the formation is transient, even for a non-perturbed initial cloud; a
non-axisymmetric instability always develops ejecting all the angular momentum
from the scalar star. In the Proca case, by contrast, no instability is
observed and the evolutions are compatible with the formation of a spinning
Proca star. Secondly, we address the stability of an existing star, a
stationary solution of the field equations. In the scalar case, a
non-axisymmetric perturbation develops collapsing the star to a spinning black
hole. No such instability is found in the Proca case, where the star survives
large amplitude perturbations; moreover, some excited Proca stars decay to, and
remain as, fundamental states. Our analysis suggests bosonic stars have
different stability properties in the scalar/vector case, which we tentatively
relate to their toroidal/spheroidal morphology. A parallelism with
instabilities of spinning fluid stars is briefly discussed.
